at 
saa ae ee 


Mere 
tag ag eee 
' ©. 


Er, : 
ig Ae ths, 


ont aes 


. * 


“a . 2 S823 


age 


Sine, 


| Be POPPE 


2 
W 
ee 
ral 
= 
5 
ea 
an 


OF ILLINOIS 
LIBRARY 


 Reammoeeeed. 


PRES He ST he 2 


Spi nt th me 


eeediqneresinlelinin ehnsna sere omtina, <meta ames htt mbit nie 


A 


ane 


py, 


Poe tee wee ee 


FIELD ENGINEERING. 


A HAND-BOOK OF THE 


THEORY anp PRACTICE or RAILWAY SURVEYING, 
LOCATION, AND CONSTRUCTION, 


DESIGNED FOR THE 


CLASS-ROOM, FIELD, AND OFFICE, 
A LARGH NUMBER OF USHFUL TABLES, 
ORIGINAL AND SELECTED. 


BY 


WILLIAM H. SEARLES, C.E., 


MoOMBER AMERICAN SOCIETY OF CIVIL ENGINEERS. 


SIXTEENTH EDITION, 


PWEEE TH THOUSAND, 


NEW YORK: 
JOHN WILEY & SONS, PUBLISHERS, 
53 East TENTH STREET, 
1895. 


CopyRiGcut, 1880, 


By JoHN WILEY & 50Ns. 


Press of J. J. Little & Co. 
Astor Place, New York. 


PREFACE. 


AuLTHouGH the modern railway system is but about fifty 
vears old, yet its growth has been so rapid, and the progress 
in the science of railway construction so great, as to render the 
earlier technical books on this subject inadequate to the needs 
of the engineer of to-day. 


In the course of his practical experience as a railway engi- - 


neer, the author was strongly impressed with the want of a 
more complete hand-book for field use, and finally concluded, 
at the solicitation of his friends, to undertake the preparation 
of the present volume. 

The aim in this work has been: 

First—To present the general subject of reilway field work 
in a progressive and logical order, for the benefit of beginners. 

Second—To classify the various problems presented, so that 
they may be readily referred to. 

Third—To embrace discussions of all the more important 
practical questions while avoiding matters non-essential. 

Fourth—To employ throughout the work a uniform and 
systematic notation, easily understood and remembered, so 
that after one perusal the formule may be intelligible at a 
glance wherever referred to. 

Fifth—To express the resulting formula of every problem 
in the'shape best adapted to convenient numerical compu- 
tation. 

Siath—To furnish a large variety of useful tables, more com- 
plete and extended than any heretofore published, especially 
adapted to the wants of the field engineer. 

An elementary knowledge of algebra, geometry and trigono- 
metry on the part of the reader has been taken for granted, as 
a command of these instrumentalities is deemed essential to 
the education of the civil engineer. The few references to 
mechanics, analytical geometry, optics and the calculus may 
be assumed correct by those not conversant with these 


DAN Say 


branches. 


iV PREFACE. 


Many of the problems in curves are new, yet there is hardly 
one that has not presented itself to the author in the course of 
his practice. The investigation of the valvoid curve is original, 
and though the mathematical discussion is somewhat difficult, 
yet the resulting formule, taken in connection with Table X, 
are exceedingly simple and convenient for the solution of a 
certain class of problems. 

The treatment of compound curves is novel and exhaustive. 
A few general equations are established, which, by slight 
modifications, solve all the problems that can occur. 

No discussion of reversed curves is given, because these are 
inconsistent with good practice, except in turnouts, under 
which head they are noticed. 

The chapter on levelling includes a discussion of stadia 
measurements, with practical formule. The chapter on earth- 
work contains a review of several methods for calcuiating 
quantities, and states the conditions under which these suc- 
ceed or fail in giving correct results. 

Among the tables, numbers 3, 5, 6, 10, 18, 19, 26 and 29 
are original. The adoption of versed sines and external 
secants throughout the work, wherever these would simplify 
the formule, rendered necessary the preparation of tables of 
these functions. The table of logarithmic versed sines and 
external secants has been computed from ten-place logarithmic 
tables of sines and tangents, so that the last decimai is to be 
relied on, and no pains have been spared to make the table 
thoroughly accurate. 

Tables numbers 4, 7, 8, 9, 11, 12, 13, 14 and 86 have been 
recalculated, enlarged, and some of them carried to more deci- 
mal places than similar tables heretofore published. The 
intention has been to give one more decimal than usual, so that 
in any combination of figures the result of calculatiop might 
be reliable to the last figure usually required. 

The tables which have been’ compiled and rearranged are 
numbers 1, 2, 15, 16, 17, 24, 25 and 81. The tables of log. 
sines and tangents here given are the only six-place tables 
which give the differences correctly for seconds. The table 
of logarithms of numbers is accompanied by a complete table 
of proportional parts, which greatly facilitates interpolation 
for the fifth and sixth figures. 

In all the tables, whether new or old, scrupulous care has 


PREFACE. 


been taken to make the last figure correct, and the greatest 
diligence has been exercised by various checks and compari- 
sons to eliminate every error. It is, therefore, hoped and 
believed that a very high degree of accuracy has been ob- 
tained, and that these tables will be found to stand second to 
none in this respect. 

The preparation of this work has extended over several 
years, as time could be spared to it from other engagements. 
It is, therefore, the expression of deliberate thought, based on 
experience, and as such is submitted to the judgment of 
brother engineers. If it shall prove to have even partially 
met the aim herein announced, and so shall serve to smooth 
the way of the ambitious student, or to assist the expert in his 
responsible duties, the labors of the author will not have been 
in vain. Wa. H. SEARLES, C.E. 
New Yorg, March ist, 1880, 


TABLES. 


I. Geometrical Propositions.........--....- Bate ete rc on 74 

Il. Trigonometrical Formule@.........--.+seses eee ee cece ereeee 275 
Aiba Curve MOrnati laa: ssh ecreee tare creer lets enero ee otain se Vei ierele ae QU¢ 
IV. Radii, Offsets, and Ordinates............. arate weet ose wien 280 

V. Corrections for Tangents and Externals..........-.......-. 288 

VI. Tangents and Externals to a One-Degree Curve............ 289 
VII. Long Chords and Actual Arcs..........-..-20 esters eee eeeee 993 
VIII. Middle Ordinates to Long Chords...... ......------..--+.+s 298 
TX.’ Linear Deflectiong. 3 coe vos ec cle o ea oem owen vice eis vem oslo 301 
X. Curved Defiections; Valvoid Arcs..........-.-.-2+-+--e-00-- 308 

XI. Frog Angles and Switches................. : pita wae eieelebaoeee 303 
XII. Middle Ordinates for Rails. ........060+-+00-e ens see cee ee es 304 
XIII. Difference in Elevation of Rails. ......-.....--.---+-----+8- 304 
XIV.) Grades and ‘Grade Angles te ercc cairns ste cee oat ete ere 305 
XV. Barometric Heights, in feet..... igies en Facets sees een ie eee 307 
XVI. Coefficient of Correction for Atmospheric Temperature.... 399 
XVII. Correction for Earth’s Curvature and Refraction........... 309 
XVIII. Coefficient for Reducing Stadia Measurements............. 310 
XIX. Logarithmic Coefficients, Stadia Measurements............ 811 
xX. (Lengths of CirctlarAreg tere. ce -cne sss eee ee mais emai 8312 
XXI. Minutes in Decimals of a Degree......... ....- SS. fs eee 313 
XXII. Inches in Decimals Of a. Hoots. ou: iran ee eine ors oot 814 
XXIII. Squares, Cubes, Roots and Reciprocals.... .-...-..-.. «- 815 
XXIV~ Logarithms of NUMDGrs ees <tc ee ene oe Serer 832 
XXV. Logarithmic Sines, Cosines, Tangents, and Cotangents..... 359 
XXVI. Logarithmic Versed Sines, and External Secants........... 404 
XX VII. Natural Sinésiand Cosimesin ease = i-tac 2 siete cto se ereernelote 25. 49 
XXVIII. Natural Tangents and Cotangents. ..........--.++++-eeeeseee 458 
XXIX. Natural Versed Sines and External Secants......-........- 40 
XXX. Cubic Yards per 100 Feet in Level Prismoids............+..- 493 

XXXI. Useful Numbers and Formule............ (Gah de TARGA DIN oats 


CITAPPER’ I. 


RECONNUISSANCE. 
SECTION PAGE 
2. Topographical considerations. .........--.ceeseeeteeenees voeeseee 1 
Be Use. OF WORDS oso gec vase Pee sghl cas soos ts See nes es Sem cae aee 3 
Pe POCKeu MStTUMMCMUS. oe a2. as oe alec + atte etarroniionce wiaaietet tthe ouernre crs sievers 3 
0 ay ASRS) 00 be i er ene Ra rattan Reon ieee incident aici’ OM AE Cobs9 + FDIC 4 
10. Pormulss fOr AMET ose a, fs a sae sieveponel oy staraya; a cl ane arama eke pate eae bebeaeich ae 4 
TG TOCKE AO VE) Sy raise ce eres eis also oie ee oof oae ratenth cmmven ata es ts Seale o ss Stage suesaza ets 7 
TZ” PrismMatiG: COMPASS <5 ie sjcccse evs ccy Sve oles o,siciere + eee OU Anes s Seis ple 7 

CRAVPTER II. 

PRELIMINARY SURVEY. 
Le DTI EL ORIG She Neh ees aie ete Te ie Rcted era erat oicconmclolctel obese arn’ oensrenrers 8 
19. Engineer COTps..-...... sce e cece cece cece rec ewcccsnesreceneccereees 8 
Oh Ciel Sarina ys. eS ene theo se pein ahbieln was Sal's simwiataTe bio! 9a ete 8 
D1 ASSIStANt OM IMCS ss cclser » clesie o acclee oo oltre, vinteiaitra dala nieie alaisle sieimcls\tiel8's 9 
| De CAV ALININV LTS Se ee tae ooh aoe cca Tork via we clorale is el ole ci oRMeKeebeatate miomaue sate he's ole 10 
| AAAS OUITALT Se ere 5 Nate ie etre Una o nian. eens el atelarerndaretevh oly arakel's skater ote a 11 
! S62 TOPORTAPNSL Actos dsc te es ccbras cog be usens vet pmaetparaes sien wees sear 11 
| EP Tres VOL ee eater et ra tee sae eaten a Sis o a'el eat ems a eis ene eerie laters wigs Marcie e's 13 
| ys ROTC Oa fee te lees AE oe bn cb Ne UP RE he Bes Pin a wha een ss 3 
| Fa FRG COMPASS £.c/s des caog dol se se ot hag Ga cums Reet Ree ees ote etree te 14 
) EMITS EL ULTA sc tovay tare a ane aroha ea as ah aos Pavol algcat eh re sembeta atatas kere fe etenar at ey avetel erecta ntote 14 
; BRE Love 06 55 cus eS) ea ONT ar ee RY cee eeeae ch Baas (Oko 
SRA THER TOGS alc otc ns aha ceria 4 aisle: © ses bisja! elass aes raretO piel olerpcis Gime tanoue’: ocetersiertys 15 
SR THE CHMOMELET So... vj, sjeree.0 ojtuie ober ose welelont oie tekegeinte nee cierto ogal fol ge nis 16 
| BE Te tlAanG-tAble. ccwsceec-s0vpi sa ca >: sae SURE a nea sini te pt 16 
| AUTO UEATISID A tte: clesalotein cis abelelsie snc Ee te c34 he ee a ee 16 
| APM TAN SI DOWES cco sie seccmie etn 2, ea Oise cero el apreastn nee sie iceleseminiaieieisis 17 
| AS es Trangit Gage ast. os 2 haan ols dae orng aieee renee ofeen inset Ue 
| 44, Obstacles to alignment and zneasurement....... Sane Mery: 18 
Ae Parallel nes... on. ne co elsccle pallet are ep ole are esialeiann syercle aisha zirpneie’s 18 
| 46. Lines ata smatl angle. ... ..). 20: .o005 5 on Beeler ad cies wsloe m ne.si6 18 
| AY, General problem... .. 2... a0 105 eae om sede snes anise noe ole cle Hees ann els 19 
| 48, Lines at a large Angle... ......- newer cece r etn rce estes meee cers 20 
49, Selection of angleS:....... 2.2 secececcee ces seer e ete rene ese cceneee 21 
| 58. Rocky shores; Tie-limes.... 2... .ceseeee eee s eee e reese cere eeeecece 22 
. System of plotting MapP......-. cece ee eee eee rete eeees Sy oes 23 


CONTENTS. 


CHAPTER III. 


THEORY OF Maximum EconoMy IN GRADES AND CURVES, 


SECTION PAGED 

Bor Cnoice Of YOULES. .. « .2.s sa snip eee ae alee nicks 6 gee earn tere 24 
“Statement ofthe problems. 2... epee tse sie > os eee eels es 2 

Engine traction .....202. 55 deca tes Sten eae g= 4 sees es aac 25 
~ Engine expense.....6.. 2065 oe Ses mele oo eens + Seer nee ems + eee 26 
GQ erd Bu atchtci Fe OLE Om ROW £00615 (0)s PERO G SS hES SOME SROese < spon a sonnaaek 27 
Resistance’due'to grade =.: Saeet cs see ce > eee ne ele ie eles 20 
2) Resistance: AUC LO Cul VC tea e wit eee ere eat eine bie mio alas meted om gel 28 
© Rormuls for resistances: e-ee bee anc clini oe arene lee sieetnwe=ir's 28 
. Formule for maximum trains........... as pac ani cheese 29 
Pan PIne=StaLek-\: nse eer ieee ee Peat ate GS eae sicko ee 30 
6. Graphical solution...... wef tee see et wc eee Sve ER nets oat 2 
. Train -load-reciprocals =: ag. 5. -elied. a. rte ne « 2s ese sclee olteey mes ae gee 
. Reduction-of grades ON CULVES..... 2.52.6 cee ieee eee eee eee eee 33 
J) Behe OS ome ac otc pugs PE ge 2 ROE RES Une aioe Ahise 51 wan Se teleee 33 
Assisting engines; 2.1ccicc etic dettole Smt ones elt ott ne oie = erate reel 36 

71. Maximum return grades: ... 2... 0.2.8 ee ss eee eee eet cere neon eeaee 36 
WD. Undulating grades 1.11.21). ahi desieiiekicistet = «sialon eraieieta siete estan te 7 
75. Comparison Of routes... 2.0... ..e cece cece rece cece tee ceneee ceceres 38 
VED Value Of distanGe:SAaVCG 2. . cee cies ocr caee ci eo a = aisle weieinseiniclgis oat 39 
PO CONCLUSION Pos. ss seve bole cee mee eee Oe is ae oe eo oreo rsa spas tats 39 

CHAPTER IV. 
LOCATION. 
pCleneral-TeCmarks. d2eccs- caesar oe ee Rm id avace eoreberets 39 
BY Longe tangents... . 2+. 05 ses smdesinees Wes ses sim aes ais Eis Sista = SO 
lueveller’s duties; Profiling ss: = sees see MEGS en Sos ti eee 
. Establishing grade lines..........--..:---.----+-=- et Ores! 
CHAPTER. V. 
SIMPLE CURVES. 
A, Elementary Relations. 

82. Limits to curves and tangents: ....... 6...6..5.: 5 a Bate eaters sae 4)} 
33) Definition of terms e.5.-.-e eee Cie tates teeta sets Cee MM ASARLSNE 45) 
84. Radius and degree of curve..... Ss SEDER Ns 3s patios NRE LS 43 
65>. Measurement of curves....... hie se WH oe bichata ta a Sees wie Ute ARs lols chet Panos. thee 44 
30. Approximate value-of B...2. 28.22% Aki we wists bine bis tie ee ey Malet eaters 44 
87. Central angle and length of Curve,........- -.s-eeseee eee e eee eeee 45 
83. Definition of other elements’. 2. Gaescce.se ss arr RRs Ae Ree 46 
99, Formula for tangent distance: D7... 2s. cuts emcees oes nee senie's 46 
ois Kormula for lone chord C@s22: st. escs. 2: oo SPP AER HOST 47 
92? Formula for middle ordinate M: 2252: co.cc dsc svee eee esos see's 48 
93. Formula for external distance #:..2:. 22.55. cence cece cee ct ecsses 48 
95. Formula for radius in terms of Tand A...............- SC ae 49 
95. Formula for external distanze in terms of Tand A............. 50 
07. Formula for radius in terms of Hand A............ oat, JE) eee Oe 


CONTENTS. 


SECTION PAGE 
98. Formula for tangent distance in terms of Hand A...... ...... 51 
99.: To. define the curve ofan old tracks. ..e ete secre cece es 51 

10)2 Other curvesformulay Panis ES See eee cures coos rime 52 

B. Location of Curves by Deflection Angles. 

j[0s Pal BYsva (axeynteyaWt: 4 s faq (s\-jaeeeeD Mrs eR POM RA COT AC CAC ore tage ar” De 

dies Ble Or AENeChONS jasc. stereeie nice 5 Scola tee mene loses en slee fase spei 8 ke = rr 53 
103, Rule for finding direction of tangent at any poijnt.. ............ 53 
RU LO CHRONOS oc ierod aha Since ohio one eae lee ay ti sls sore en fae Oe 
105. WeHeCLLONS 1 OF SUDCNOLGS 4 -.. sijcc «4 sey ois eirpeiaie eet ee biel teh ie hate 54 
JOGr Correchions fOr SiO CMOTOS a 5 ora jor ysis o osaie et sreue ake iets eee pes eaiejenseln anes chit 55 
ip RaMOMOL COLTEChION UOrOXCESS OL OTC. oases aclaielorcve slatenietnictet-ieieusiieeis = 55 
POS Treut Site WOM KOT MOUS VCS ones, creek dw) ans ao, wigs odes lao nll oan carded Ae aT 57 
PRO OIC IO LES secs 9 oe. peta xea wie = 2,0 snia. euoiehche dig elena siete eee ale are eae te 58 
110. Central angle in terms of deflections....... 00... .-.d meee +. 6-2 oe 58 
fii Method Dydenleohons On] Vice .t-yo.0.0i5 « stencil eae eualre eta ae 58 

C. Location of Curves by Offsets. 

1127 Rourmethods seek - ee eae eee ee aera Nike Gaels 59 
113. By offsets from the chords produced............+--s.seeeeeeee eee 59 
11 Doshec inning witty a SUDCHOLGS aoc tart. settee tere lelatel a ter-taen 60 
115. Formula for subchord offsets, approximate....... ST eC 61 
117. By middle ordinates... seas os soe 0 ies 2h s Seiden anwcas saree e eyes 61 
118. Do. beginning with a subchord.......-....-----.ee cece eee eee ees 62 
DLO B yaar Ont Ori SUS ares where bos Netto Sets Stole sroreicing inte inlays seat an ela wiehcas 62 
120. Do. beginning with a subchord........ 2.26... - see eee eee cece recone 64 
12t-By ordinates fromalong chord >... .).° 7s eee nee oe 64 
1298 Dosfor‘an even number of stations... 10" 5. ae essa cee ores scree ee 65 
1233) Por foran odd number Of Stations. .--c.o. ce cette te aceis ese ses carce 66 
124, Do. for an even number of half stations......... ..........-2.-.. 67 
125. Do. beginning with any subchord............-..---..--eeseee sees 67 
126. Erecting perpendiculars without instrument.................-.-. G9 

D. Obstacles to the Location of Curves. 

Tota ENE VELUOX IN ACCESSIOL Gs. 2.5. so aba sae aaiatne catia pers Cole teicher syste el lot 69 
128. The point of curve inaccessible...........-- 2.2.2 ee s eee se ne 7 
129. The.vertex and point of curve inaccessible.................----. 70 
130, The point of tangent inaccessible... ............-- 0, -0++++-+-+-ees fi 
131. TO pass. an Obstacle Ona CUIVE: © 6.2 aise cecilia oe? ln eine 2-2 = 72 

EE. Special Problems in Simple Curves. 

132. To find the change in R and # for a given change in T........... (¢ 
133, To find the change in R and T for a given change in #........... 74 
134. To find the change in T and E for a given change in f........-.. 79 
135. General expression for elementary ratios...............+-----0- 75 
136. To find a new point of curve for a parallel tangent.............. 76 
137. To find a new radius for a parallel tangent...................... 76 


38 To find new P.C. and new radius for a parallel tangent......... 


SECTION 
139, 


CONTENTS. 


To find new tangent points for two parallel tangents............ %8 


140. To tind new R and P.C. for new tangent at same P.T............ 80 
141. To find new P.C. for a new tangent from same vertex........... 81 
142, To find new radius for a new tangent from same vertex......... 81 
143. To find new R and P.C. for same external distance, butnew A. 82 
144, To find a curve to pass through a given point............... aactars 83 
145. To find new radius for a given radial offset.....................- 84 
146, equation Of the -velVvOld: tees werretemle ter ee oie cca ale ae are re 86 
147. To find direction of a tangent to the valvoid at any point........ 7 
148. To find the radius of curvature of the valvoid at any point...... 88 
149. To find the length of are of the valvoid......... IS fas a's deep eres 88 
150. To find new position of any stake for a newradius from same 

Pie ee eae ee re aT ee tree eo 89 
151. To find new radius from same P.C. for new position of any 

batho sic sh fe 26 os Ee eae eee ee area oe enn deo loa caatee ote eters 92 
152. To find distance on any line between tangent and curve, ........ 93 
158. To find a tangent to pass through a distant point................ 94 
jot, To finda line tangent LOU WOlCUrVES as teen. ee bs hie een 96. 
155. To find a line tangent to two curves reversed..............2.020 98 
156. Study of location on preliminary map; Templets; Table of con- 

Venient .CUIVES ised cle be EER ee ieee via Irckats fevalectociaenes Rta e ees 100 

CHAPTER VI. 
CoMPOUND CURVES. 
A, Theory of Compound Curves, 

i. WDETIDITION . jtesti is acc bons Sace.a.0 DSc Sree Rae ees PA elt Se eee 102 
158. The circumscribing circle............... adie Gal pterete Mapes ERT ET 102 
159. The locus of the point of compound Curve.. ©. .¢.0 52.0. <s0%8 ve <0 103 
160. The inscribed circle of the principal radii.......0.....c.0..00.00- 104 

Cor, 2, Maxima and minima,.o£ thewadiia.) sic ae sesiesee ee 104 
B, General Equations. 
161. Formula for radii, central angles, and sides..................... 105 
162" Given: S,.S,° 4 and Rystotind)7 wee ese and ee ee 106 
163. Given: AB, VAB, VBA. and R,, tofind A, A,and R,.. ..... 107 
164. Given: Rk, A, R, Ag, to find the triangle VAB. oo. 0 .. otee ene 108 
165. Given: A, the radii, and one side to find the other... ........-. 108 
166. Given: one side, radius and central angle to find the others..... 110 
o/s Remarks on special Casesijzeex .5c secke cee sche oie eae eee 111 
163 Obstacles; the.P.C.C.anaccessiblece. 6 ua. hte eee eee 112 
: C. Special Problems in Compound Curves. 
169; Todind a new P.C.C. fora, parallel tangent...,.........-..-?sseens 1138 
170, To find a new P.C.C, and last radius for a parallel tangent...... 115 
171. To find a new P.C.C. and last radius for the same tangent...... 118 
172. To find a new P.C.C. and last radius R.’ for new direction of 


tangent throuch sameP. 7) ever ers. en a tlie ete ene aie ot 


4) 


CONTENTS. 


SECTION PAGE 
173. To find a new P.C.C. and last radius FR,’ for new direction of 
tangent through same P.T.......6. eee eee eee e eee eens agiad Wa! 
174. To replace a simple curve by a three- centred compound curve 
between the same tangent points.........-6-- esses eee ee eee 127 
175. To find the distance between the middle points of a simple curve 
and three-centred compound CUrVe......-. ss cece eee cee tees 129 
176. To replace a simple curve by a three-centred compound curve 
passing through the same middle point........--.---+- ee ie 
I. The curve flattened at the tangents.........--.... eee seer eee 129 
Il. The curve sharpened at the tangents. ....-...---4.-++++++eees 1382 
177. To replace a tangent by a curve compounded with the adjacent 
CUILEV OR SL ah rae tee crt EE tie dee Pistons rareeevateraiene sieve atalee 134 
I. When the perpendicular offset p is assumed.........----+-+ 136 
II. When the angle q or Bp 1S*ASSUIMEG ie ee Crete a iclees caine 137 
TIl. When the radius R, is assumed ............- ee eee eee eee BT 
LVes Locus-of the. centre Oj ah ikica ee et sate reece mate ergiaierer=tel 133 
178. To replace the middle arc of a three-centred compound by an 
Are Of Ciftorent TAGING .. 6 so. 8 bees ce cepemaincle ewe otic ae orcs 140 
I. When the radius of the middle arc is the greatest,.......... 140 
Il. When the radius of the middle are is the least.............. 141 
III. When the radius of the middle arc is intermediate.......... 142 
CHAPTER VII. 
TURNOUTS. 

9. Definitions; Frogs and switches.....:....---+-+seseee sees ece cree 147 
180. Single turnout from straight track in terms Of from angle ...... 148 
181. Single turnout from straight track in terms of frog number..... 149 
182. Double turnout, middle track straight, to calemate Ht... am 151 


183. Double turnout, middle track straight, and three given frogs... 152 
184. Double turnout on same side of straight track to calculate the 


middie frogs Hie esse ach See Ses conir eaeec city te aut elas 153 
185. Double turnout on same side of straight track with three given 
PTOGS 0 ok hone pe sink ote reek eaas os manga asia scl 2p haa ca oie 155 
a. When the middle track is a simple curve.....-.-++-+-++++++++ 155 
_ When the middle track is straight beyond F’.........--+-.+-- 158 
. When the middle track is reversed at P.......+--+-++--++-+++5 159 
186. scout on the inside of a curved track.......-..-.+--+0+ seeeee- 161 
187. Turnout on the outside of a curved track....... ..++-+++-+-+0s> 163 
188. Tougue switches 1.2... opens sawinny uoriitegasAgment meme hems ¢ 164 
189. Tongue switch turnout from a straight track.......:.2.0sseeee- 164 
190. Tongue switch double turnout to GheQded ah (pte ee wc odo y ooo enor 165 
191. Tongue switch double turnout with three given frogs..........-. 166 
192. Tongue switch double turnout on same side of straight track 
with three given frogs........--+----+++-- eS Ben SMA hors 167 
a. The middle track reversed at F........--.-- 22s cee cece ences 167 
b. The middle track compounded at F’........-.--- Etat & 168 


c. The middle track straight beyond I’.... po Ne ared age aes (ola: 


x11 CONTENTS. 


SECTION PAGE 

193. To find the reversed curve for parallel siding in terms of F' and 
perpendicular: distance sass peeee aa ea eee oe +3, 169 

194. To find the connecting curve from frog to parallel siding on a 
curve in terms of #'and perpendicular distance p.......... 176 
a“The siding: outside of main track cf aseiltts dhe bese 171 
H b-2The siding inside, of mainntrack..ny isewwdcaee eS eee ity 171 
! 195. To locate a crossing between parallel tracks. ..:...........0.26- 17% 
! 186. To locate a reversed curve crossing between straight tracks.... 178 
197. To locate a reversed curve crossing between curved tracks..... 174 
198. To find the middle ordinate m, for one station in terms of D.... 175 
i 199. To find the middle ordinate m1 for rails, in terms of rail and R.. 1% 
200, Curving rails; To find m1 in terms of railand m................. 176 
201, To find elevation of outer rail On CUrveS « s...6. 6.6) ieee el ee 177 

202. To find a chord whose middle ordinate equals the proper eleva- 
ANON. Ses os aes 6 ois See eo OIE ee a eee ee ee 179 
203, General remarks on elevation, of Tail! eaiec ws tone eek ee 19 
204, General remarks on coned: wheels............ sc.eseceee seeees . 180 

CHAPTER. VItt 
LEV£L1LI.G. 
205. Use of the engineers’ level. ... 2.2.0.0... c.ccee cece cecees eee ots 181 
DUOe Le datum, how ASSUNIECc susseincas aes oe ode ce see sbislde eee ee eine 181 
Avie benchés: how Used $2 BW eee, faa ee acl a Sea 181 
Ave. FL eie ht, Of  Tstrumrent steal see ee Oe ole os ae aes ee 182 
209 -Reading Of the Trodscs bes eee eee ee 6 cae owe oe ae eee 182 
210, -Llevatiom- of mtermediate points: << +.2.s2 . 5a. thawinsheee aeneee 182 
Bed at ACT SOLUTE See) See Sar stiatewiert ie Se neo eR eee 182 
Pie. KUle tor DACkKsights and t OPEsi@nts om. -..0 acetate cen Sere 183 
mis, Horm of field-book: proofot extensions. ......cc.-..2. oe 183 
ber OE VONIGRY 2. ws. a> 2 oo 07 we'd apo ROR TEE Ub atineeGlt DE ERIE One ee 1&4 
mio. mibrple leveling; test levels... n1s. need. eset eet sce foe tare 185 
216. Errors in reading, due to the level; how avoided.=.............. 185 
217. Errors in reading, due to the rod;. how avoided.................. 185 
mis; hors GUC LO CUrVALUTe Of THEIeALb hate ee. sete: oe oe See eer 186 
mig Barors due to refraction -4 > soso. ee nee be es chs Saltese s orate ee 187 
220.) Wadlus OF Ciryature Of the Garhiits. cose seks seater eee 187 
Eels evelling by Lransit:or theodolllewunmetcs «nectes cee eee 188 
22, To find the HJ. by observation of the horizon................... 189 
223. Stadia measurements; horizontal sights......................... 191 
224, Stadia measurements; inclined sights, vertical rod.............. 193 
225, Stadia measurements; inclined sights, inclined rod............. 195 
j 
CHAPTER 1X 
CONSTRUCTION. 

e260. Organization of engineer departments. 2s... 65sul,4-0s Jessie... 196 
Reve Hearine and Tub bing a5 4 Sk Amsaeherccee ae Ot ae eee 197 


p Lest levelsand:enard. phipgs acu keene at eee ee ae ete ee! 


CONTENTS. 


SECTION PAGE 
990 OrOSs SECHONS* “SlOPOSe si. 22 1s ole nie,0 oe Soleo as Se eiettnieencistare oss wea, hod: 
230. Cross sections, formule for........ SEE AIG HO Coe Ses Sent ane aie $8 
931. Cross sections, Staking Out... . 2.2... ccs seccwsre ee Bias anise 200 
232. Cross sections on irregular ground..... aire he BRS iss oe por Ree 201 
238. Cross sections on side-hill work.................+--- sgh 3 AOC 201 
234, Compound Cross sections.............04. AEE PCRS AOL E ree foe 202 
235. Selection of points for cross sections............ SEaset nes ahha 203 
DEG)  WOEiICAl, CUEN ES > oe cieia aisle. << apelan oegsuane oalatobenene piel avatoysLoleltarieln ateastayspet 2 203 
O37 HOT OF CLOSS-SCCLLOM DOOK® =~ clas n.d cele siere sietetate ere erate elataal=)s08\aiatelal a/2 204 
O38: ixtended Cross profiles. .i32 yt) eatiac grasa «seietet-/-i=10 Meteors cl 205 
BIO TNACCESSIDIO SOCLIODS « «5 = ais shoe ccyo is ots al eta citielebs eles ato ers Samed iocniatet © 205 
DANS LSGLADCC TVA SSCS Aa akils.ccc 2 x e-t.c:w «) cue) oro wich teeid etavatel ene tortie eiehah ance csacartes oie 206 
Ste BOLLOW -pitecnia se yc cleeis.s e's te Lae SS yas aa ee Soca re ea eae ots eis 206 
POS Sarin kare selIICrEAS’.. sie os «dese dors ol evslela eleieyaiel suet Melee talc) aeaetelete eteiaior= 206 
WARPED ECE VWoORICS « c scels hase 2 Seas ose 3. pore one A ee te olen eet era ee 20% 
OAT A ILETAtIOMIOL UTIG., ..c.cr- a ciazeiccie sone elapoteierpie) tie memes ererers ret sams, otras ene 207 
945 DrainS and Culverts, « <o<.c a. <sjslald -/aa:oeiavelethe Alar alberta Set 2,o ya easke is anaes 208 
DAG Se ACG Mertsn scans: os ae bicisis shail aero emer taerre Utes g dake Ce eee 209 
247, Foundation pits; Bridge chords on curves.........--.++++-+--0+5 210 
O49 Cattle-OUBrds Jo 4 6 cbs ses e-eiee 10:0 + mom nen e.8 bee eee Ral aoa eet a 214 
DAG IT POSEIG=WOLKK 25 a clcisd < sieve are cise: os -sisbare/e Bi eienege lel Nos Lape emee teeth Ws atiel yn aistay tak . 214 
250. Tunnels:. Location; Alignment; Shafts; Curves ; Levels ; 

Grades; Sections; Rate of progress; Ventilation; Drain- 

DIES we cieieie e Late 3 a's] nseieis s/sceie jae « ewisn'e sees ae cleieink= ciarey> /e siahiiestammaats 216 
951. Retracing the lin€.......... ec cee eee cee eee n ee ett teers ey eee: 222 
959. Side ditches and. drains. 5... x. 2. e106 sisls «slay over starperers]© wimrsoPaererreags 223 
BES TBallastine so. Feo. ais owe evo Seis tie owinitin ¢ a HN mele Siale 01s mewteresiialeaates Sainte 223 
254. Track-laying; Expansion of rails; Sidings...........-..-+-++++-- 223 

CHAPTER X. 
CALCULATION OF HARTHWORK. 

954. Prismoids: Choice of cross sections. .v... 0... 26. coer eee eee eee 225 
£55. Formule for sectional areas.........-- .--.-+----+-+++6-- tS atone 220 
256. Prismoidal formule for solid contents. .....-......+--.+---0e-es- 229 
257. Tables of quantities in cubic yards... .....-...---.e eee eee eee ee 229 
258. Tables of equivalent depths............ 2... cece eee e eee teen eee. 231 
259, Formula for equivalent depth in terms of slope angle........... 268 
260, Conditions necessary for correct results in use of tables........ 25: 
261. Method of mean areas; correction required... ...............06-. Qee 
262. Exact calculation of content; examples.................. Shite eins ROO 
263. Wedges and pyramids. .........0. 22 cece eee acetic ees cece een 286 
264. Side-hill sections, uniform Slope. ....... 0.522... .050 veecesee eee 226 
265. Side-hill sections, irregular ground..............--.---+- PS ey 237 
266. Side-hill secticns in terms of slope angle.............-....-,---- R230 
267. Systems of diagrams. ...... 6... ccsasene sue dsines se snes ewes wes nobus 238 
268. Correction for curvature in earthwork................-.222-. eee 239 
269. Haul; Centre of gravity of prismoid........ 2.2.2... ee ee eee ee 243 
BiO> Pinal OSUIMALO qca.5 5 clo oe os Lee a ah air ak Waa ae eo wets Gar are ee 245 
271. Monthly estimates. ,......... rage rages esate ra 246 


CONTENTS. 


CHAPTER XI. 


TOPOGRAPHICAL SKETCHING. 


SECTION PAGE 
ac General remarks 0.0.5. 000 2 aie ae a eka en 247 
gi) ACU Acial foatures. <. . i, a eee eas ee Ome blag 248 
274, Natural features; Contours; Hatchinegs. 2.1.7. , 3) EP GE ORS, » 248 
275. Method of sketching -.. 1g aos te tye een. eee 249 
CHAPTER XII. 
ADJUSTMENT OF INSTRUMENTS, 
eco The transit ....:.-.. 22a aeataan saan enclmeeanne Aiea katie 250 
Bec abe level... :.2. 7241.5. diame Aeepeaieuet rg cena eae 252 
gOS Tie theodolite. :-:2-2f5: ots asian 1 aeirad a een 2538 


CHAPTER XIII. 
EXPLANATION OF THE TABLES, 


wine gy tt FO eh ster ahera! eteNeteratela tates Siete’ Tee cok Ree EN tee 


FIELD ENGINEERING. 


CHAPTER I. 
RECONNOISSANCE. 


1. The engineering operations requisite to and preceding the 

construction of a railroad are in general: 
THE RECONNOISSANCE, 
THE PRELIMINARY SURVEY, and 
THE Location. 

2. The Reconnoissance-is a general and somewhat hasty 
examination of the country through which the proposed road 
is to pass, for the purpose of noting its more prominent 
features, and acquiring a general knowledge of its topography 
with reference to the selection of a suitable route. The 
judicious selection of a route may be a very simple or com- 
plex problem, depending on the character of the topography, 
and more especially on the direction of the streams and ridges 
as compared with the general direction of the proposed road. 
2. A road running along a water-course is most easily 
located, In this case the choice is to be made merely between 
the two banks of the stream, or between keeping one bank 
continuously and making occasional crossings. When the 
stream is small it will usually be found best to cross it at 
intervals, the advantage of direct alignement outweighing the 
cost of bridging; but when the stream is of considerable size 
the solution of the problem is not so obvious, requiring patient 
comparison of results in the two cases to determine whether to 
cross or not, while in the case of the larger rivers crossing 
may be out of the question. 

When there is a choice of sides, both banks should be 
traversed by the engineer on reconnoissance, and while exam- 
ining in detail the one side he should take a general and com- 
prehensive view of the other. Only thus can he gain a complete 
knowledge of either side. The points to be considered are the 
relative value of the property on either side, the number and 


FIELD ENGINEERING. 


size of tributary streams, and probable cost of crossing them, 
the cost of graduation as affected by the amount and character 
of the material to be removed, and the liability to land slides, 
the amount and degree of curvature required, and the proba- 
ple revenues which the road can command. If, in respect to 
these points, one bank of the stream gives the more favorable 
result all the way, the question is decided at once; but in 
case the greater inducements are found on either bank alter- 
nately, as usually happens, the propriety of bridging the 
stream, with the costs and advantages, must be considered as 
an additional element in the problem. 
4. When no water-course offers along which the road may 
be located, the difficulties of selecting a route are increased, 
and these usually become greatest when the streams are found 
to run about at right angles to the direction of the road. Val- 
leys and ridges are to be crossed alternately, involving the 
necessity of ascending and descending grades, diverting the 
road from a straight line, and increasing the distance and cur- 
vature. The engineer must now seek the lowest points on the 
ridges, and the highest banks at the stream crossings, in order 
to reduce as much as possible the total rise and fall, but these 
points must be so chosen relatively to each other as to admit 
of their being connected by a grade not exceeding the maxi- 
mum which may be allowable. The intervening country 
between summit and stream must usually be carefully exam- 
ined, even on reconnoissance, to determine where the assumed 
grade will find sustaining ground at a reasonable expense for 
graduation and right of way. 

In selecting stream crossings, regard should be had not only 
to the height of the bank, but also to the character of the bot- 
tom, its suitability for foundations, and its liability to be 
washed by the current. The direction and force of the cur- 
rent should be observed, and its behavior during freshets, and 
the extremes of high and low water ascertained, if possible. 
An approximate estimate of the cost of bridging may be made, 

5. The engineer should not only seek the best ground on the 
route first assumed, but should have an eye to all other possi 
ble routes, holding them in consideration pending his accu- 
mulation of evidence, and being ready, finally, to adopt that 
one which promises the greatest ultimate economy. He should 

be able to read the face of the country like a map, and to 


RECON NOISSANCR,. 3 


carry in his mind a continuous idea or image of any line he is ex- 
amining, so as to judge with tolerable accuracy of the influence 
any one portion of the line may have on another as to align. 
ment and grade, even though many iniles apart. In the success- 
ful prosecution of a reconnoissance he must depend mainly on 
his own natural tact and a judgment matured by experience. 

G6. The engineer will bring to his aid in the first place the 
most reliable maps, and those drawn on the largest scale. The 
sectional maps of United States surveys will be found very 
useful when they exist. In addition to these it is often desira- 
ble to prepare a map on a scale of one or two inches to a mile, 
on which will be drawn the principal features of the country 
to be traversed, such as streams, roads, towns, and the princi- 
pal ridges, if known, but leaving the further details to be filled 
in by the engineer as he progresses. Such a map furnishes a cor- 
rect scale for his sketches, and saves much valuable time, as he 
has only to sketch what the map does not contain, and occa- 
sionally to make corrections when he finds the map to be in 
error. He also notes on the map the governing points of the 
route, such as the best crossings of streams, ridges, or other 
roads, and any point where the line will evidently be com- 
pelled to pass. He may then indicate the route by a dotted 
line on the map drawn through the governing points. Having 
traversed the route in one direction he should retrace luis steps, 
verifying or correcting his observations, and making such 
further notes as seem important. When in a densely wooded 
country, with but few openings, it may be impossible for him 
to get a commanding view from any point that will afford him 
the necessary information as to the general topography. He 
must then depend largely upon instrumental observations, 
taking these more frequently, and noting carefully all details 
likely to prove useful in future surveys. 

¢. The instruments required on an extended recon- 
“noissance are the barometer and thermometer, the hand or 
Locke level, a pocket or prismatic compass, and a telescope or 
strong field-glass. To these may be added a telemeter for 
measuring distances at sight, but when good maps are to be 
had this instrument is seldom needed. o also some portable 
astronomical instruments are necessary in a new country, for 
determining latitude and longitude, but would only be a use- 
Jess incumbrance in a settled district, 


FIELD ENGINEERING. 


$. The mercurial barometer has generally been relied upon 
for the determination of heights, but owing to its inconvenient 
dimensions and the danger of breaking, it is now discarded by 
railroad engineers in favor of the more portable aneroid 
barometer, except in the case of trans-continental surveys, 
and when astronomical instruments are to be used also. 

9. The best aneroids are designed to be self compen- 
sating for temperature, so that with a constant atmospheric 
pressure the reading shall be the same at all temperatures of the 
instrument. This, however, being a very delicate adjustment, 
is not always successfully made, so that each instrument is la- 
ble to have a small error due to temperature peculiar to itself. 
This error will be found rarely to exceed one hundredth of an 
inch, plus or minus, per change of ten degrees Fah., and is 
frequently much less than this. Just what the error is in a 
particular instrument may be determined by careful compari- 
son with a standard mercurial barometer at the extremes of 
temperature, assuming the error found as proportional to the 
difference of temperature for all intermediate degrees of heat. 
The error having been determined for any aneroid, it should 
be applied, with its proper sign, to every reading to obtain 
the true reading. 

The sizes generally used are 13 and 24 inches in diameter, 
respectively, and experience seems to prove that there is no 
advantage in using larger sizes, but rather the contrary. 

10. The ordinary barometric formule and tables have been 
prepared with reference to the mercurial barometer. In order 
that they may apply to the aneroid, it is necessary that the 
latter should be adjusted to read inches of mercury identically 
with the mercurial column at the sea level at a temperature of 
39° Fah. But as the aneroid, unlike the mercurial column, 
requires no correction for latitude, nor for the variation in the 
force of gravity due to elevation, that portion of the formula 
which provides for such corrections, as well as that which 
provides for a correction due to the temperature of the 
‘instrument itself, may be omitted when using an aneroid. 
Thus the general formula is very much simplified, and be 
comes 


z= log Hf 60384.3 (1 + a 64 ) 


RECON NOISSANCE. 5 


in which i, and 7’ are the readings of the aneroid in inches, 
and ¢, and @ the readings of a Fahrenheit thermometer at the 
lower and upper of any two stations respectively, and z is the 
difference in elevation in English feet of those stations. 
To facilitate the calculation of heights by this formula, we 
may write 
Log . 60384.3 = [log 2, — log h'] 60384.3 


and since only the difference of the logs. is required, this will 
not be affected, if we subtract unity from each. The quan- 
tities in Table XV. are prepared, therefore, by the formula 
(log 2 — 1) 60884.3 
for every ;2,ths of an inch from 19 inches to 31 inches. 
+ t’ — 64° 
~ 900 


Table XVI. contains values of f for every de- 
gree of (¢, + 1’) from 20° to 200° Fah. 

11. To find the difference in elevation of any two stations by 
the tables : 

Take the difference of the quantities corresponding to /, and 
h' in Table XY. as an approximation, and for a correction 
multiply this difference by the coefficient corresponding to 
(¢, +2, in Table XVI., adding or subtracting the product 
according to the sign of the coefficient, 


Hrample.— 
Lower Sta. Upper Sta. 
in. in. 
Aneroid h, = 29.92 h' = 23.57 
Thermometer i tne lien tg 8 ai 
By Table XV. for 29.92 we have 28741 
for 23.57 22485 


Difference 6256 
By Table XVI. for 77.6 + 70.4 = 148 we have -+ .0983 


Then 6256 X .09383 = 583.6848 
and 6256 + 584 = 6840 ft.= 2.—Ans. 


12. Certain precautions are to be observed in the use of the 
aneroid. When the index has been adjusted to a correct 
reading by means of the screw at its back, it should not be 
meddled with until it can again be compared with a standard 
mercurial barometer, and even then some engineers prefer to 
take note of its error, if any, rather than disturb the aneroid. 


§ FIELD ENGINEERING. 


Again, since the principle of compensation supposes the 
aneroid to have a uniform temperature throughout its parts, it 
must be guarded against sudden changes, as otherwise the 
metallic case will be considerably heated or cooled before the 
change can affect the inner chamber, thus inducing very erro- 
neous results. ‘The aneroid, therefore, should seldom be taken 
from its leather case, nor exposed to any radiant heat of sun 
or fire, nor worn so near the person as to increase its tempera- 
ture above that of the surrounding atmosphere. If removed 
to an atmosphere of decidedly different temperature, time 
must be allowed for the aneroid to be thoroughly permeated 
by the new degree of heat. The aneroid should be held with 
the face horizontal while being read; it should be handled care- 
fully, and all concussions avoided, and it should be compared 


with a standard as often as practicable to make sure that it 


has suffered no derangement. Observing these precautions, 
and having a really good aneroid, the engineer should obtain 
excellent results in the estimation of heights. It has been 
found that the slight error in compensation, previously alluded 
to, is subject to a change during the first year or two after the 
instrument is made, but subsequently it becomes quite per- 
manent. 

13. For the purpose of obtaining approximate elevations by 
a simple inspection of the dial, the modern aneroid is provided 
with a secondary scale reading hundreds of feet, which is 
placed outside the scale of inches. It is divided according to 
the following formula prepared by Prof. Airy: 


é h,—l' t+t— =) 
Eo MAD core ‘ ; 
i 550382 i (1 = 1000 


in which it is evident that no correction for temperature is 
required when the average temperature of the two stations is 
50°. When the two scales are engraved on the same plate the 
zero of the scale of feet is coincident with 31 on the scale of 
inches; but in some aneroids the scales are on two concentric 
plates, so that the zero of one may be made to coincide with 
any division of the other, which is in some respects an advan- 
tage. 

14. The theory of the barometer, as expressed in the above 
formule, assumes the atmosphere to be at rest, and its pres 
ure affected only by temperature, whereas, in fact, the pres: 


RECON NOISS.A NCE. 7 


sure at any point is liable to sudden changes due to variations 
in the force of the wind, the amount of humidity, etc. The 
best way to eliminate errors due to these causes is to take read- 
ings simultaneously at the points the elevations of which are 
to be compared. For this purpose an assistant should be 
stationed at some point of known elevation contiguous to the 
route to be surveyed, and provided with an aneroid similar to 
that carried by the engineer. The aneroids, time-pieces, and 
thermometers having been compared at this point, the assist- 
ant should record the readings every ten minutes, with the 
time, temperature, and state of the weather. The engineer 
will thus have a standard with which to compare his own 
observations. If the survey is so extended that the same con- 
ditions of atmosphere are not. likely to be experienced by the 
two observers, the assistant should be instructed to move for- 
ward to a new station at a designated time; or two assistants 
may be employed, one at each of two stations between which 
the engineer intends to make a reconnoissance. Even with 
these precautions no attempt should be made to obtain the ele- 
ration of important points during, or just before, or after a 
storm of wind or rain. 

15. When but one aneroid is used the observations at the 
several stations should be taken as nearly together as possible 
in point of time, and then repeated in inverse order, taking 
the mean of the observations at each station, and repeating the 
whole operation if necessary. Only approximate results can 
be hoped for, however, with a single instrument, unless the 
atmospheric conditions are very favorable. 

16. The Locke Level is an instrument in which the 
bubble and the observed object may be seen at the same instant, 
enabling the operator to keep the instrument horizontal, while 
holding it in the hand, like an ordinary spy-glass. While 
very portable, it enables the observer to define rapidly all visi- 
ble points of the same elevation as his own, and_to estimate 
from these the relative heights of other points. It may be 
made useful in a variety of ways which easily suggest them- 
selves to the engineer in cases where no great precision 18 
required, and where a more elaborate level is not at hand. 

17. The Prismatic Compass is a portable instrument 
with folding sights, in using which the bearing to an object 
may be read at the same instant that the object is observed. 


8 FIELD ENGINEERING. 


The bearings are read upon a floating card, graduated and 
numbered from zero to 360°, so that no error can be made in 
substituting one quadrant for another. The instrument may be 


held freely in the hand during an observation, though better 


results are obtained by giving it a firm rest. 


CHAPTER II. 
PRELIMINARY SURVEY. 


18. A preliminary survey consists in an instrumental exam- 
ination of the country along the proposed route, for the 
purpose of obtaining such details of distances, elevations, 
topography, etc., as may be necessary to prepare a map and 
profile of the route, make an approximate estimate of the cost 
of constructing the road, and furnish the data from which to 
definitely locate the line should the route be adopted. The 
survey is more or less elaborate, according to circumstances. 
In case the country is new, or the reconnoissance has been 
incomplete, or if several routes seem to offer almost equal 
inducements, the survey will partake somewhat of the nature 
of a reconnoissance, and will be made more hastily than if but 
one route is to be examined, and that, perhaps, presenting 
serious engineering difficulties. The survey is made as expe- 
ditiously as possible, consistent with general accuracy, but 
should not usually be delayed for the sake of precision in 
matters of minor detail. 

19. For preliminary survey the Corps of engineers is 
organized as follows: 

A chief engineer, an assistant engineer, two chainmen, one 
or two axemen, a stakeman, and. a topographer, these forming 
the compass (or transit) party, to which a flagman is some- 
times added; a leveller and one or two rodmen, forming the 
level party; and to these is sometimes added a cross level party 
of two or three assistant rodmen. 

20. The chief engineer takes command of the corps, 
and directs the survey. He ascertains or estimates the value 
of the lands passed over, the owners’ names, and the boundary 
lines crossed by the line of survey, He examines all streams, 


PRELIMINARY SURVEY. 9 


and estimates the size and character of the culverts and 
bridges which they will require; he notices existing bridges, 
and inquires concerning their liability to be carried away by 
freshet; he selects suitable sites for bridges, examines the 
character of the foundations, the direction of the current rela- 
tively to that of the line, and considers any probable change 
in the direction of the current during freshets; he’ inspects the 
various soils, rocks, and kinds of timber as they are met with, 
and takes full notes of all these and kindred items in his field 
book. He not unfrequently assumes in addition the duties of 
topographer. He should run his line as nearly as may be over 
the ground likely to be chosen for location, so that the infor- 
mation obtained may be pertinent, and so that the length of 
the line, the shape of the profile, and the estimate based on 
the survey may approximate to those of the proposed location. 
To thisend he has due regard to the levels taken, and when 
they show that the line as run fails to be consistent with 
allowable grades, he either orders the corps back to some 
proper point to begin a new line, or makes an offset at once 
to a better position, or continues the same line with some 
deflection, simply noting the position and probable elevation 
of better ground, as in his judgment he thinks best. He 
should at all times maintain a friendly attitude toward pro- 
prietors, and by his polite bearing endeavor to secure their 
cordial support. of the new enterprise. If he is tolerably cer- 
tain that the location will follow nearly the line of the prelim- 
inary survey, he should have with him some blank deeds of 
right of way, and let these be signed by land-owners while 
they are favorably disposed. When this cannot be done, ¢ 
blank form of agreement to allow the surveys and construc- 
tion of the road to proceed until such time as the terms of 
right of way may be agreed upon may be made very useful. 
The chief also selects quarters for his men, and in case of 
camping out he directs the movements of the camp equipage. 

21. The assistant engineer takes the bearings of the 
courses run, and makes a minute of them, with their lengths, or 


the numbers of the stations where they terminate. He sees that. 


the axemen keep in line while clearing, and the chainmen 
while measuring; he takes the bearings of the principal roads 
and streams, and of property lines when met with. In an 
open country he may save time by selecting some prominent 


10 FIELD ENGINEERING. 


distant object toward which the chainmen measure without his 
assistance, while he goes forward and prepares to take the 
bearing of the course beyond. In traversing a forest with not 
too dense undergrowth, when the line is being run to suit the 
ground according to a given grade, it is a good plan for the 
assistant to go ahead of the chainmen as far as he can be seen, 
select his ground, take his bearing by backsight on the last 
station, and then have the chainmen measure toward him. In 
this case both he and the head chainman should be provided 
with a good sized red and white flag, mounted on a straight 
pole, to be waved at first to call attention, and afterward held 
vertically for alignement. Otherwise a flagman must be added 
to the party, who will select the ground ahead, under the in- 
structions of the chief, and toward whom the survey will pro- 
ceed in the usual manner. 

22, The head chainman drags the chain, and carries a 
flag which is put into line at the end of each chain length by the 
assistant engineer or the rear chainman. It is his duty to 
know that his flag isin line and that his chain is straight and 
horizontal before making any measurement, and to*show the 
stakeman where each stake isto be driven. <A stake is usually 
driven at the end of each measured chain length, called a 
station, though in an open and level country the stakes at the 
odd stations may be omitted, in which case marking pins are 
used to indicate the odd stations temporarily. In case there 
is much clearing to be done the head chainman plants his flag 
in line, and ranging past it, indicates to the axemen what is 
to be cut, going a little in advance through the bushes so that 
they may work toward him. The head chainman should be a 
quick, active and strong man, with a good eye and a taste for 
his work, as very much of the real progress of the survey 
depends upon him. 

23. The rear chainman holds his end of the chain firm- 
ly at the last stake or pin by his own strength, not by means of 
the stake. He keeps the tally by the pins when they are used, 
and watches the numbers on the stakes to see that they are cor- 
rect. The end of a course should always be chosen at the end 
of achain, if possible, and if not, then at a brass tag indicating 
tens of feet, as thus the labor of plotting the map will be much 
lessened. The numbering of stations is not recommenced 
with each new course, but is continued from the beginning to 


PRELIMINARY SURVEY. 1] 


the end of the survey, through all its courses, and if one 
course ends with a portion of a chain the next course begins 
with the remainder of it. It is. the rear chainman’s duty to 
attend_to this, holding the proper link at the compass station, 
Any fraction of a chain measured on the line is called a plus, 
and is counted in feet from the previous station. The length 
of an offset in the line is never included in the length of the 
line, but if the line should change its course by a right angle, 
or more, or less, the numbering would go on as usual. 

24. The axemen should be accustomed to chopping and 
clearing, and are, therefore, to be selected in the country rather 
than the city. They will cut out so much of the underbrush 
and overhanging branches as may interfere with the sight of 
the assistant or leveller; but care must be taken not to cut 
unnecessarily wide, and no tree of considerable size should be 
felled, except in rare instances. When running by compass, if 
the assistant goes ahead of the chain, he can always select a 
position so that no large tree will interfere; or, if the line must 
be produced and strikes a tree, the compass may be brought up 
and set close to the tree on the forward side as nearly in line as 
can be estimated, the slight error in offset being neglected, 
since the lin2 will b2 produced parallel to itself by the needle. 

25. The stakeman prepares and marks the stakes, and 
drives them at the points indicated by the head chainman. 


When no clearing is needed, the axemen keep him supplied 


with stakes, as the rapid progress of the chain will only give 
him time to drive them. The stakes should be two feet long and 
pointed evenly so as to drive straight, and are blazed or faced 
on two opposite sides, one of which is marked in red chalk 
with the number of the station. The stake must be driven 
vertically, and with the marked face to the rear, so that it may 
be read by the rodman as he follows the line. 

26. The topographer makes accurate s<xetches of all 
features of the country immediately on the line, and extends 
the sketches as tar each side of the line as he can, in a book 
prepared for the purpose. He must never sketch in advance 
of the chain, nor in advance of his own position. His work 
should be done to scale as nearly as possible, using the same 
scale for distances on the line and at right angles to it. The 
scale adopted should never be less than that of the map to be 
made from the sketches. The ruled lines of a field book are 


12 FIELD ENGINEERING. 


usually one quarter of an inch apart, so that a scale of one 
line to a station equals about four hundred feet to an inch. 
This is the smallest scale ever used. The scale of two lines to 
a station is most convenient for general use. Four lines to a 
station are needed in special cases to show details, as in pass- 
ing through villages. The scale may be changed from time to 
time as found necessary, but no two scales should ever be used 
on the same page. The numbering of the stations up the page 
indicates the scale of the sketch. 

27. When the contours of the surface are required, the 
topographer may join the level party in order that his esti- 
mates of heights and slopes may be corrected by the instru- 
ment. He should never draw a mass of contours indiscrimi- 
nately, but should sketch them as they exist at a uniform ver- 
tical interval. This interval may be assumed at five feet in 
a gently rolling country, and at twenty feet in a mountainous 
one, but an interval of ten feet will be found most convenient 
generally. If the topographer accompanies the level he can 
assume the contours at the even tens of feet in elevation, and 
mark them so, noting where a contour crosses the surveyed 
line, and sketching its direction and shape both ways from 
that point. He will estimate the rate of slope of the ground 
at right angles to the line as so many feet per hundred, and 
record it from time to time, noting ascent from the line on 
either side by ‘‘ A,” and descent by ‘‘D.” If the slope changes 
within the limit of the page, the line of change may be 
sketched and the next slope recorded. When little banks or 
terraces occur, or bDluffs and rocks, which cannot be suf- 
ficiently indicated by contours, they should be shown by 
hatchings, and the height noted. Special care should be 
taken to sketch roads and houses in their correct positions 
and dimensions, the latter to be either measured, paced or 
estimated. The dimensions should also be recorded in num- 
bers. The outline of forests may be shown by a scalloped 
line, and the kind of timber, and whether dense or scattered, 
written within the inclosed space. Correct outlines are essen- 
tial, but no time should be given to shading up a sketch with 
conventional signs. A single sign, or the name of the thing 
intended, is all sufficient. Land-owners’ and residents’ names 
should be recorded whenever they can be obtained, as well as 


the names of roads, streams and public buildings. 


4 


PRELIMINARY SURVEY. 13 


28. The leveller takes charge of the level party and 
keeps the notes of his work. He reads the rod on benches and 
at turning points to hundredths of a foot and to tenths at other 
points. He should direct a bench to be made at least once 
every half mile, and in a very rough country every quarter of 
1. mile. The benches need not be far from the line, and, if 
well chosen, may be used as turning points, thus saving time. 


i 
The elevation of turning points must be computed when 


tuken, so that the elevation of any one of them may be 
instantly given when called for, and the other elevations will 
be filled in as far as may be without delaying the survey. As 
the levels are usually the most essential part of the survey, 
much care should be taken to have the instrument well ad- 
justed and truly level, and the rod held vertically and correctly 
read on turning points, but the intermediate work should not 
be so done as to delay the party unnecessarily. The leveller 
should use every endeavor to follow closely after the survey- 
ing party, so that the chief and topographer may have the 
advantage of his notes. 

29. The rodman’s first duty is to hold the rod vertically, 
and he must learn to do this in calm or windy weather, in level 
field or on side hill. He may carry a small disk-level, which 
applied to the edge of the rod will show when it is vertical. 
The turning points are to be selected for firmness and definite- 
ness, and so that they will afford a clear view from beyond 
for a backsight. The rod is held for a reading on the ground 
at every stake, the number of which is called out to the level- 
ler as soon as the rodman arrives at it; the rod is also to be 
held at every prominent change of slope on the line, as the 
crest and foot of every bank, the rodman calling out its dis- 
tance from the last stake as plus so many feet, but all gentle 
undulations and minor irregularities are to be neglected. The 
rod will always be read at the surface of a stream or pond, 
and also at its deepest part on the line, when possible; other- 
wise the depth of the water may be found by sounding, and 
so recorded. Should the line run along a stream the surface 
will be taken occasionally, opposite certain stations, and in 
case of a canal, the elevation of surface above and below cach 
lock must be noted. The rodman makes inquiry for high- 
water marks or seeks traces of them himself in an uninhabited 
district, and holds the rod upon them that their elevation may 


14 FIELD ENGINEERING. 


be determined. The rodman carries a small axe or hatchet 
with which to make benches and to trim out any stray 
branch that may intercept the leveller’s view. 

80. The assistant rodmen take the slope and elevations 
of the ground at right angles to the line, using vertical and hori- 
zontal rods and a pocket level, or a tape line and clinometer. 
The cross levels are not taken throughout the whole survey, if 
at all, but only where the roughness of the country seems to 
demand it. They may be extended to any distance from the 
centre line required by the chief—not less, however, than fifty 
feet as a rule. They may be taken at the stations only, or 
oftener, if necessary, depending upon the roughness of the 
surface, the object being to define accurately the contours, 
and so the shape of the ground. The assistant rodmen will 
also take soundings when they are needed, either on the line 
or at right angles to it. 

31. In defining the duties of the members of the corps, the 
instruments used have been incidentally noticed. 

32. The compass is preferable to the transit on prelimi- 
nary surveys, because it can be operated more rapidly, is lighter, 
and usually has a better needle. It may have either plain 
sights or telescope, and be mounted on tripod or jacob staff. 
The simpler forms are preferred for forest work. Not unfre- 
quently the engincer’s transit is employed, but using the needle. 
A preliminary line should not be run by backsights and deflec- 
tions, unless local attraction is found to exist to such an extent 
as to destroy confidence in the needle; or, in special cases, 
where the natural obstacles to a survey are very great. In the 
latter case the survey partakes of the nature of a location, and 
should be conducted with similar care and fidelity. 

33. The chain is 100 feet long, and composed of 100 
links. It should be of steel for lightness, durability, and greater 
accuracy. Those having rings of hard stecl, unbrazed, are 
least apt to wear. Five marking pins are needed, each having 
a piece of red flannel attached, for temporary stations, or for 
keeping points temporarily while measuring by parts of a 
chain up or down aslope. A pointed plumb bob, with sev- 
eral yards of small cord wound on a carpenter’s spool, is use- 
ful in chaining over steep declivities or bluffs. 

34. The axes should be of. best quality, with hand-made 
handles, and not too heavy. The axe of the stakeman should 


\ 


PRELIMINARY SURVEY. 15 


have a fine edge for dressing and a broad head for driving 
the stakes. When the stakes are not required to be over two 
feet long, a stout basket, having.a square, flat bottom, 26x14 
inches, should be furnished the stakeman. He will then pre- 
pare a basketful of stakes, ready marked, and place them in 
the basket regularly, in the reverse order of their numbers, so 
that they will come to hand as wanted. A small hand-saw 
no larger than the basket, with rather coarse teeth, wide sct, 
will be found extremely useful in cutting stakes with square 
heads and of uniform length, and much more rapidly than can 
be done with an axe. When not in use, it is to be strapped to 
the inside of the basket, to prevent its being lost by the way. 
When the basket is about empty, the stakeman, with the 
assistance of the axemen, can soon replenish it, and the stakes 
being all numbered at once, there is less danger of a mistake 
being made in the tally than when they are marked only as 
wanted. 

35. The level should be the regular engineer’s level, the 
same as used on location. 

306. The rod should be self-reading, ¢.¢., to be read by the 
leveller, as too much time would be consumed in the constant 
adjustment of a target by the rodman. It should be as long as 
can be conveniently handied in order to reduce the number of 
turning points on hill sides. A very convenient rod may be 
made of thoroughly seasoned clear white pine, sixteen feet 
long and two inches wide, with a thickness of one inch at the 
bottom, increasing to one and a quarter inches at six feet from 
the bottom, and then gradually diminishing to three eighths of 
an inch at the top. The rodis shod with a stout strap of steel, 
extending five inches up the edges, and secured by screws. 
The top is protected for a few inches by a plate of sheet brass 
on the back. The face of the rod is a plain surface through- 
out, and is graduated from the lower edge of the steel shoe as 
zero. The divisions are fine cuts made with the point of a 
knife. At the foot and half-foot points the cuts extend across 
the face. For the tenths and half tenths they extend three 
quarters of an inch from the right hand edge, terminating in a 
line scribed parallel to the edge of the rod, thus forming rec. 
tangular blocks half a tenth wide, every other one of which is 
painted black, the body of the rod being white. The feet are 
indicated by numerals painted red on the blank part of the 


SS iE 


16 FIELD ENGINEERING. 


face, each figure standing exactly on its foot mark, and being 
exactly one tenth high. For the figure 5 the Roman. \V. is sub- 
stituted and for 9 the Roman I[X., so that in case a dumpy 
level is used the 5 may not be mistaken for a 3, nor the 9 fora 
6. At the half-foot points a red diamond is painted, so that 
the graduated line bisects it. No other figures nor gradua- 
tions are required. With this rod the leveller can read quite 
accurately to hundredths of a foot, and after some practice 
can estimate the half hundredths. 

o%¢. The horizontal rod for cross-levels may be made 
of white pine, ten feet long and one inch thick by three wide, 
tipped with brass, painted white, and graduated to feet and 
tenths. It must be a straight edge, and is levelled by a pocket 
level placed upon it when needed, or by a small level embedded 
permanently in one edge. The vertical rod to be used with it 
is made of pine eight feet long and one and a quarter inches 
square, and graduated to feet, tenths, and half tenths. All 
rods when not in use should be laid on a flat surface to pre- 
vent their being sprung. Leaning them in acorner soon ruins 
them for use. . 

28. The clinometer is any small instrument which will 
measure the slope angle of the surface. The angle is always 
estimated from the horizon, a vertical being 90°. The rise per 
100 feet is found by multiplying the nat. tangent of the slope 
angle by 100. It may often be found more easily by the 
leveller reading the rod at a station and then 100 feet left or 
right of the line. If surface measures are taken in connec- 
tion with a.slope angle they are reduced -to horizontal meas- 
ures by multiplying them by the cosine of the slope angle. 

39. The plane-table is rarely if ever used on prelimi- 
nary surveys in the United States. Occasional bearings taken 
to prominent objects by the assistant engineer, or the use of a 
prismatic compass by the topographer in connection with his 
sketches, is found to answer every purpose. 

40. In case a survey is to be made with a tran- 
sit, it is necessary to add a back flagman to the party, who will 
hold his flag or rod on the point last occupied by the transit, so 
that the assistant may take a backsight upon it. The direction 
of a new course in each case is determined by the deflection 
angle to the right or left of the preceding course produced. 
The bearing of one long course near the beginning of the sur- 


PRELIMINARY SURVEY. 17 


vey having been carefully ascertained, the bearing of each suc- 

ceeding course is calculated from the BeHestiona, and entered 
in acolumn of the field book headed Culeulated Beart vngs, from 
which the line is afterwards plotted. The magnetic bearing 
of each course should also be taken from the needle, and re- 
corded as such, but is used only as a check on the transit 
work. The deflections should be made in degrees, halves, or 
quarters, if possible, to facilitate the calculation of bearings, 
and to admit of using a traverse table. 

44. The attached level and vertical arc of the transit are 
useful in determining approximately the grade of the line run 
in advance of the level party, or in seeking for one assumed 
grade to which it is desired that the line shall conform. For 
this purpose it is only necessary to set the vertical arc to the 
angle corresponding to the grade as given in Table XIV., and 
let the head chainman move about until a point on his rod at 
the same height from the ground as the telescope is covered 
by the horizontal cross-hair. 

42. The point on the ground where a transit is set up is 
marked by a good-sized plug, flat headed, and driven down 
flush, with the ground, with a tack set in the head to show the 
exact point or centre. This is called a transit point. When 
a transit point occurs at a regular stetion, the stake bearing 
the number of that station is set three feet to the left cf the 
jine opposite the plug and facing it. When a transit point 
occurs between stations the stake is driven three feet to the 
left of it, marked with the number of the preceding station 
+ the distance from that station in feet. 

43. As a transit is capable of giving a line with great pre- 
cision, it is important that the flags used in connection with 
it should be equally precise in giving points, An excellent flag 
for this purpose is made of well-seasoned clear white pine ten 
feet long, two and a half inches wide, and one inch thick. It is 
tapered for the last four inches to an edge at one end, the edge 
being formed at the middle of the width, The tapered end is 
shod with a band of steel covering the edge of the rod, and 
secured by screws, and the steel is brought to a sharp edge at 
the point of the rod. The rod is then painted white and 
tipped with brass at the square or upper end, A centre line 
on the face is then struck from the point of the steel to the 


18 FIELD ENGINEERING. 


middle of the brass tip by means of a piece of sewing silk, 
and a fine cut made with a knife and steel straight edge. 
The centre line must not be scribed parallel to one edge of the 
rod, as this is rarely ever straight. The face of the rod is 
then divided into one-foot spaces, measured from the head of 
the rod, and these are painted red on either side of the centre 
line in alternate blocks. On the back of the rod at three and 
a half feet from the point is placed a small ground-glass 
bubble-tube, mounted very simply, and attached to the rod by 
a brass plate and screws, and guarded by two blocks of wood 
for protection. The centre line of the rod is made vertical by 
a plumb-line while the level tube is being attached, which ever 
after secures a vertical rod. If only two feet of this rod can 
be seen over any obstruction, a point can be set with great 
| precision, provided the level tube is in adjustment. This flag 
| can also be used as a plumb in chaining with much more 
satisfaction than a cord and weight, especially in windy 
weather. 

44, A transit survey usually requires more clearing than 
one made by compass. When a given course is to be produced 
in a forest, some large trees will inevitably be encountered, but 
the labor and delay of felling them may be avoided hy the 
use of auxiliary lines. These may be classified as running 
parallel to the main line, at a small angle with it, or at a large 
angle with it. 

45. The parallel line is established by means of two 
short perpendicular offsets measured with care before reach- 
ing the obstacle, and the main line is established beyond the 
obstacle by means of two more equal offsets, But since short 

back-sights are to be avoided, these offsets should be at least 
i 100 feet apart, so that it may be difficult to find a parallel line 
of sufficient length which does not strike some other obstacle, 
1 || or at least require considerable extra clearing. 

46. The auxiliary lines making a small angle 
with the main line are more convenient, not only on this 
account, but because they require a less number of transit 
points. By them an isosceles triangle is formed on the ground, 
having the intercepted portion of the main line as base, and the 
vertex near the obstacle. The deflections at the points where 

the lines leave and join the main line are similar and equal, and 


bh 


PRELIMINARY SURVEY. 14 


the deflection at the vertex is double in amount and opposite 
in direction. No calculation is necessary, for the error in 
measurement due to the deviation is too small to be noticed. 
and since the main line is immediately resumed, the calculated 
bearings of the auxiliary lines are unnecessary. Should. the 
point where the second line joins the main line prove unsuit 
able for a transit point, the second line may be produced to 
any convenient point beyond, and so go to form an isosceles 
triangle on the opposite side of the main line, the triangle 
being completed by running a third line parallel to the first, 
and equal to the difference of the first and second. Again, 
the second line may encounter a serious obstacle before reach- 
ing the main line. To avoid this a parallel to the main line 
may be run from the end of the first line for a con- 
venient distance, and there the second line be put in 
parallel and equal to its first position, as before de- 
scribed, thus forming a trapezoid. 

47. The following general solution of this 
problem allows the engineer to make use of. any 
number of auxiliary lines, provided that none of 
them make an angle of much more than one degree 
with the main line, with a certainty of resuming the 
main line in position and direction at the extremity 
of any course desired, and without necessitating 
any trigonometrical calculation. It is based on the 
assumption, practically true for small angles, that 
the sines are proportional to their angles, and is ex- 
pressed by the following rule: 

Call all deflections to the right plus, and all to 
the left menus ; multiply the length of each course 
in feet by the algebraic sum in minutes of all the 
auxiliary deflections made to reach that course; 
take the algebraic sum of these products, and 
when the sum equals zero the extremity of the last 
course will be on the main line. The deflection 
required at that point to give the direction of the main line 
is equal to the algebraic sum of all the preceding deflections, 
but taken with the contrary sign. 

Thus, if we have left the main line at A, and run by these 
notes: (Fig. 1.) 


Fie. 1. 


FIELD ENGINEERING. 


Sta- Defi. Dist. Factors. Products. 
A. 16° R 199 2% +16 X 190 = -+- 3040 
B 31 L 1200 $3. — 15 x 120= — 1800 
C 18’ R 175 E 5 te BX 1D D0 | 
D 13' L Le Desai 10 X 265 = = 2650 
E 15’ R Ba + 5X (?) 

3065 — 4450 
and their algebraic sum is ~ — 88d 


Therefore to render the sum zero we must add 885 as the pro- 
duct of the last course. But 5’ is already given as one factor, 


eld 


885 » gent 
so that the other factor must be ae 177, which is the length 


of the last course, giving some point /’ on the main line. The 
deflection at / from the last course to give the direction of 
the main line is 


16 — 31-4 18-1384 15=95' 


and changing the sign we have — 5’; that is, the deflection is 
to the left. 

The distance on the main line from A to / equals the sum 
of the courses, or 927 feet, but this we have by the stations, 
which have been kept by stakes in the ordinary way. All the 
stakes on the auxiliary lines will be more or less off the main 
line, but as these offsets are usually very small, they are con- 
sidered of no consequence on a preliminary survey through a 
forest. In Fig. 1 the offsets are very much magnified. The 
field notes of such auxiliary courses should be Kept, not as 
regular notes, but on the margin or opposite page, and in such 
a way that, while the line may be retraced by them on the 
ground, the draughtsman may see that it is not necessary to plot 
them, when a straight line raled and measured through is suf- 
ficient. It is obvious that in selecting a closing course either 
the deflection may be assumed and the length calculated, or 
vice versa ,; but care should be taken to assume such values as 
do not involve a fraction in either factor, if possible. 

48. The method of passing an obstacle on the line by 
auxiliary lines at a large angle with the main line will 
only be resorted to when circumstances are such that the other 
methods mentioned cannot be employed, as in passing a build- 
ing, pond, or densely wooded swamp. In such a case we may 


PRELIMINARY SURVEY. 21 


turn a right angle with the transit, and measure accurately one 
offset, putting a transit point at its extremity, where another 
right angle will give a parallel line. If the offset prove too short 
for an accurate backsight, a temporary point at a sufficient 
distance may be established for that purpose on the offset line 
produced before the instrument is removed from the main 
line. If any other angle than 90° is used it should be selected, 
when circumstances permit, so that the distances on the inter- 
cepted part of the main line may be in some simple ratio to 
the distances measured on the auxiliary line. Thus a deflection 
ot 60° gives a distance on the main line equal to half the 
length of the auxiliary course, that is, 


60° gives a ratio of += 0.5 

HB IS EH bi es $ 0.6 nearly 
45° 344" us UT AES 
mad are es s 0:55 8e 
25° 50y Shs e 0:9 gars 


the angles being taken to the nearest half minute. 

49. If it be desired that the stakes on the auxiliary line 
should stand on perpendiculars through the true stations 
on the main line, a certain correction must be added to each 
chain length depending on the angle which the auxiliary 
makes with the main line. If there is a fraction of the chain 
at either end of the course, a proportional addition must be 
made for this. Thus, by referring to the table of external 
secants, we find that we must add a correction as follows: 


2° 834...0.1 ft. per chain. | 6° 453’...0.7 ft. per chain. 
9 
vo 


oe De Vi a aga PAI: reiti ta 
POG AD BEST 7°.302'...09 « 
Beings i Ga iites -«s 8% (4 pee ce 
ae © a (0s PD: 6 OO 
621525 3.20-6.06 -". « TROON igi ah 


These methods of suiting the angle to an even measure are 
much superior to assuming an even number of degrees deflec- 
tion, and then calculating the distance by trigonometry. The 
last table, which may be extended indefinitely by reference to 
the table of Ex. secants, is perfectly adapted to chaining by 
surface measure on regular slopes when the slope angle is 


22 FIELD ENGINEERING. 


known, the chain being lengthened by the correction corre- 
sponding to the slope angle. 

50. If the chain is lengthened as per above table on auxil- 
iary lines, the numbering of the stakes goes on as usual, but 
they should have an additional mark as X to show that they 
are off the main line; and they may stand facing the true 
stations which they represent, and the length of offset, if 
known, may also be recorded on them. The leveller will then 
understand that he is to read the rod not only at the stakes as 
they stand, but also at the true stations, as nearly as may be. 
The assistant engineer will always make a diagram in his 
field book, showing exactly the method pursued in reference to 
auxiliary lines. Having passed the obstacle, it is advisable 
to return to the main line by a course equal in length to the 
first auxiliary, and making an equal angle with the main line. 
If this cannot be done from the end of the first course, a 
parallel to the main line may be run any convenient distance, 
and the return line then put in, forming a trapezoid. 

51. When there is no obstruction to sight on the main 
line, but only to measurement, a transit point should be 
set in line beyond the obstacle before the transit leaves the 
main line, as a check on the other operations, and the main 
line should be afterward produced from this point by back- 
sight on the main line, rather than by deflection from an 
auxiliary line. 

52. The main line should always be resumed as soon as 
practicable, making the auxiliary lines the mere exception. 
When a number of courses at a large angle are likely to be 
required before the main line can again be reached, it may be 
better to consider these as regular courses of the survey, and 
to note them as such. The sémplest method is always the dest, 
because least likely to involve mistakes. 

53. When the natural obstacles are so numerous 
and of such magnitude as to render any continuous line of sur- 
vey or location extremely difficult, if not impossible, as‘in the 
case of a bold rocky shore, all the data necessary to a location 
should be gathered with precision on the preliminary survey, 
the measurements and angles being taken with the greatest care, 
and as many checks as possible should be introduced to verify 
the work. In meandering such a shore it is probable that a large 
number of short courses will be used, which may be measured 


PRELIMINARY SURVEY. 2d 


correctly, but there is liability to error in the angles, To 
verify the latter the more conspicuous transit stations on 
prominent points of the shore are selected, and these being 
named by the letters of the alphabet, the deflections between 
them are taken by careful observations repeated a number of 
times, as for a triangulation. These points, joined by tie- 
lines, then form a survey of themselves, much simpler than 
the full traverse. To obtain the length of these tie-lines, the 
angles between them and the courses meeting at the same 
station are measured. Then since each tie-line forms the 
closing side of a field, in which all the bearings are known, 
and all the distances, save one, that one may be calculated by 
latitude and departures. But the angles should first be tested 
for error in each complete field, and if the error be large the 
angles must all be remeasured until the error is found and cor- 
rected, but if very small it may be distributed among the 
angles, or among those most probably inaccurate. Before cal- 
culating the traverse of any of these fields, it will be advanta- 
geous to assume, for an artificial meridian, a line parallel to 
the average direction of the shore for several miles, and to 
refer all courses to this meridian for their bearing. This 
meridian is called the avis of the survey, and all bearings 
referred to it are called avial bearings, as distinguished from 
magnetic bearings. The magnetic bearing of the axis should 
be some exact number of degrees, so as to facilitate the reduc- 
tion from one system to the other. 

54. In plotting the map, the axis is first laid down, and then 
the lettered stations in their respective positions, after which 
the meandering surveys can be filled in. The map being 
drawn on a scale of one hundred feet to an inch, and the con- 
tours constructed from the notes of the level and cross-level 
parties, the engineer may project the location upon it with 
great certainty and economy of result. But he should calcu- 
late the traverse of the location as projected, and compare it 
with the traverse of the preliminary, to eliminate all errors in 
drafting, before taking his notes to the field to reproduce the 
location on the ground. Any point where the location crosses 
the preliminary should have the same latitude and longitude 
by the traverse of either line. This system, though laborious, 
is the only one that will ensure a successful location under the 


circumstances supposed. Advantage may sometimes be taken 


24 FIELD ENGINEERING. 


of cold weather to cross bays and inlets on the ice, but there 
is great liability to error in angles taken upon the ice, due both 
to its motion and to the sinking of the feet of the tripod into 
the ice as soon as exposed to the rays of the sun. 


CHAPTER. IIT, 
THEeorRyY oF MAaxtmumM EconoMy IN GRADES AND CURVES. 


55. Before commencing the field work of location it de- 
volves upon the engineer to decide as to which of the surveyed 
routes shall be adopted as being most advantageous in all 
respects, and also to establish the maximum grade in each 
direction and the minimum radius of curve on that route. 

The general considerations which guide the engineer in the 
selection of one of several routes for location are such as were 
hinted at in the chapter on reconnoissance, but upon the com- 
pletion of the preliminary surveys he has at hand a large 
amount of information which enables him to consider this 
important question much more in detail. Unless his instruc- 
tions are explicitly to the contrary, he may assume it to be his 
duty to find the best line, or that one which, for a series of 
years following the completion of the road, will require the 
least annual expense, including interest on first cost. The 
finances of the company may be so limited as not to permit 
the construction of the best line at once, and it may then be 
the duty of the engineer to select the cheapest line, or that of 
least first cost, as a temporary expedient, with the expectation 
of building the road at its best when the improved credit of 
the company will permit. But generally he will be able to 
build the cheaper portions of the best line at once, only making 
deviations and introducing heavier grades at the expensive 
points to avoid a cost beyond the present means at his com- 
mand. The selection of the best line may be a question as 
between different routes or as between different grades and 
curves on the same route. We will consider the latter case 
first. 

56. To solve the problem of true economy we must 
determine the actual expense both of building and operating 


MAXIMUM ECONOMY IN GRADES, ETC. 2D 


the line at a given maximum grade, and also what changes will 
be made in these expenses by a change in that maximum. We 
have then, on one hand, the annual interest upon the original 
cost, and, on the other, the annual expense of operating the road. 
The best grade ts that which will render the sum of these two a 
minimum. Both forms or expense consist of two parts: one 
that is affected by a change in grade, and the other that is not. 
Clearly the former is the only one we have to consider in either, 
since when the sum of the variabie portions is a minimum, the 
sum total will be a minimum also. Tbe varying portions then 
are functions of the grade, though independent of each other. 
If, therefore, we let 2’ represent the maximum grade in feet 
per mile, and let @ represent the corresponding value of that 
portion of the annual expense which varies with the grade, 
and establish the relation existing between the two, we shall 
have z= f(z). Similarly if we let y represent the interest on 
so much of the first cost as is affected by grade, we shall have 
y =f' (2). The problem then is to find that value of 2’ which 
shall render 
e+y=a minimum. 


Let us now seek the complete expression represented by 
tees f te). 

The elements that enter into this expression are numerous, 
and will be considered in succession. 

57. The traction of an engine is the force with which 
it pulls a train, and is limited by the reaction of the drivers 
against the rails. It depends on the weight upon each driver, the 
number of drivers, and the coefficient of friction. The weight 
on one driver should not exceed 12,000 Ibs., and is usually less 
than this. If the exact proportions of engine that will be 
used on the road are not known, the weight per driver may 
be assumed at 10,000 Ibs., with 4 drivers for ordinary grades 
and traffic, or at 11,000 lbs. with 6 drivers, if the grades are 
steep and the tonnage large. For extraordinary grades special 
engines are required, having 8 or 10 drivers. The coefficient 
of friction, called also the adhesion, varies from .09 to .37, 
these being the extremes on record. The lowest is due to 
extremely unfavorable circumstances, as sleet and frost; the 
highest doubtless to the use of sand, though not so stated in 
the record. The more common range of values is from .15 to 


26 FIELD ENGINEERING. 


.25. For our present purpose it will be assumed at .20, se 
that if a 4-driver engine has 10,000 Ibs. on each driver, its 
traction is 40,000 * .20 = 8000 lbs. when hauling its maximum 
train. 

08. The expense of running an engine one mile, hauling 
a train, on the proposed road, can only be estimated from the 
experience on other roads similarly situated. The expense is 
composed of the Items of fuel, water, oil and waste, repairs 
(including renewals), wages of conductor, engineer, and fire- 
man, engine-house expenses, and_ interest on first cost of 
engine and engine-stall. The range and approximate average 
of these items is here given: 


| || : 
4-DRIVER ENGINE. | 4-DRIVER 6-DRIVER | 8-DRIVER 
ITEMS. || ae | 
Lowest. | Highest.'|| Average. Average.; Average. 
1 EU XE) Wied es ins eis $0.050 $0.210 $0.100 $0.165 $0.213 
Wa COT cece stems tx .001 010 || .004 006 | .008 
Oil and waste........ 904 .030 |] .006 .008 -010 
Repairs and renewals 05 150 .080 104 | 133 
WIA OS Teeter cine | 050 .100 075 075 O75 
Engine-house. ......} 025 060 .035 -050 -060 
Interest).32.00 52% aes .025 .038 030 .038 047 
OtaIs but .-e scr 205 .598 3830 A46 546 


si a= SE Pee 


In a given case the probable value of each item should be 
estimated separately, and the sum taken afterwards. In the 
above averages each.engine is supposed to haul its maximum 
train. The relative expense of the several classes of engines 
has not been established conclusively. 

59. The resistance offered to the motion of a railway 
train is occasioned by a variety of causes, concerning which 
a great deal of uncertainty exists as to their relative effect. 
An investigation which should seek to determine the exact 
amount of each partial resistance, and then by a summation 
derive the total, would be tedious, and, in the present state of 
our knowledge, unsatisfactory. We shall therefore simply 
group the resistances under three general heads, namely: 
Resistance due to uniform motion on a straight, level track; 
Resistance due to grade; 
Resistance due to curvature, 


bs 


MAXIMUM ECONOMY IN GRADES, ETC. a7 

60. The first of these, considered as an aggregate of 
the various items of friction in engine and train, of oscillations 
and impacts, and of resistance of the atmosphere, is found to 
vary nearly or quite as the square of the velocity. The frie: 
tion of an engine is greater in proportion to its weight than 
that of a car, owing to its many moving parts, so that the 
resistance of a short train is greater in proportion to its total 
weight than that of a long train. The resistance of the atmos- 


phere is greater also in proportion to the weight of a short a 
train than of a long one. An empty train will offer more Mh 


resistance in proportion to its weight than a loaded one. A 
formula which shall express the resistance of a train to uni- 
form motion must include at least the velocity and the weight 
of the train and engine. 

: The following empirical formula is based upon a careful 
investigation of all such records of experiments on the subject, 
several hundred in number, as have come to the author’s notice, 
and is believed to give results agreeing closely with the average 
experience and practice of the present day. It is designed to 
give the resistance per ton for all trains, whether freight or 


SEDO LE LOL EDEL LI 


passenger, and at any velocity, under ordinary circumstances. 
r Accidental circumstances, such as the state of the weather, 


and the condition of the road-bed, rails, and rolling stock, may 
largely modify the resistance, but these, of course, are not aa 
taken account of in the formula. ut 
Let V = velocity of train in miles per hour, | i 
“ # = weight of engine and tender in tons, AV] 
‘““ W = weight of cars in tons, i 
‘< T = weight of freight in tons, 
“q = resistance to uniform motion in Ibs. per ton. 
We then have the formula 


i 0006? 


61. The second resistance considered is that due to 
gravity in grades. It varies in the exact ratio of the rise to the 
length of the grade. 

Let G, = rise of grade in feet per station. 

‘““ G, = rise of grade in feet per mile. 
q = resistance in pounds per ton duc to grade. 


ce 


FIELD ENGINEERING. 


Then, 
. re 
q = 2240 100 4G, 
(2) 
G 14 
' — 99 a SF 
gq = 2240 5380 > 387" 


62. The third resistance considered is that due to 
curvature of the track. This resistance is due to the friction 
of the wheels upon the top of the rail, and of their flanges upon 
the side of the rail. The top friction is lateral, due to the 
oblique position of the wheel on the rail, and longitudinal, due 
to the greater length of the outer rail, since both wheels are 
rigidly attached to the axle. The flange friction is due to the 
reaction of the top friction, which, combined with the parallel- 
ism of the axles, throws the truck into an oblique position on 
the track. A forward flange presses the outer rail, while a rear 
flange is usually in contact with the inner rail. The centri- 
fugal force of the car will increase the pressure on the outer 
rail, unless the ties are inclined at an angle sufficient to coun- 
terbalance this force. But if the ties are inclined too much, 
or the velocity is less, the pressure’on the inner rail will be 
increased. An uneven track will cause the truck to pursue a 
zigzag course, increasing the resistance considerably. 

Experiments for determining the amount of curve reststance 
have been neither numerous nor very satisfactory, but the 
generally accepted conclusion is that the resistance isa little 
less than half a pound per ton on a one-degree curve, and that 
it varies as the degree of curve. On European roads, how- 
ever, it is estimated at about one pound per ton per degree of 
cv-ve, owing largely to the form of rolling stock used. 

63. Let q’ = curve resistance in pounds per ton on any 

curve, 
and D = degree of curve. ; 

Then, assuming the resistance per ton on a one-degree curve 

at 0.566, we have for any other curve 


g" = 0.56D (3) 


To ascertain what grade upon a straight line will offer the 
same resistance as a given curve; substitute the value of q’ 
for q' in eq. (2) and solve for G; whence 

G, = 9.025D 


‘4 
Ga 1 B20 ® 


MAXIMUM ECONOMY IN GRADES, ETC. 29 


For definition of degree of curve, sce Art. 84. 

GA. It is evident that grades and curves, by their resistances, 
fix a limit to the weight of a train which a given engine can 
haul over them. 

A locomotive is usually built with such a surplus of boiler 
and cylinder capacity that its power, at ordinary velocities, is 
limited by the adhesion of the drivers, so that the adhesion is 
the proper measure of the tractive force. | 

Yo find an expression for the maximum train which a gwen 
engine can haul over a given grade and curve: 

Let P = tractive force of engine in pounds, 


« 7' = weight of paying load in tons per maximum 
train, 


«« W' = weight in tons of cars carrying the load 7”. 
Then for uniform motion, at a given velocity, 


(B4+Ww+T)@+¢+o)=h (9) 


Let ¢ = average load of one car in tons 
‘w= average weight of one car and load in tons. 


‘ ae wes nae BB 
Then W’' =- 7’ = of , substituting which in eq. (5) we derive 


t P ) 
fyi So. : pS ( 
CHG, hen wt hae " 


| 


In this equation g = the resistance per ton due to uniform 
motion, g' = the resistance per ton due to the maximum grade 
opposed to the direction of the train, and q’’ = the resistance 
per ton due to the sharpest curve on that grade. 

For accelerated motion the reaction of inertia of the train 
must be added to the above resistances. This is estimated at 
4g, in order that a train starting from rest may acquire the 
requisite maximum velocity, even on a maximum grade, in a 
reasonable time, say from 8 to 6 minutes. Therefore, for 


accelerated motion, 
t tes — 
pr =+(,..-5) (7) 


of TZ and q involve each other, but if we 
eq. (1) the value of g becomes that used in 


Now, the values 
accent Wand 7’ in 


30 FIELD ENGINEERING. 


eq. (7), and we may eliminate g between these equations, and 
ilerive the value of 7’; whence 


“(P~ .00098? V7?) 5 
| F akOtals Gh age A GECDONT yee (8) 


Also, for the weight of maximum train and load, 


a — .00098? V2, 
gta +81 +0097? 


which is the expression required. 
When there is no curve on the maximum grade, g" is zero: 


| and when there is no grade, gq’ is zero; hence for a straight level 
i track eq. (7) becomes 


We 7 


—H# (9) 


and eq. (8 
Blak ey (10) 
~(P — .0009E? V2) 
(gee pe 
° =" 81+ .009 V2 w 


65. An engine-stage is a division of the road to which 
an engine is limited, and over which it regularly hauls a train. 
Its length varies, on existing roads, from 50 to 200 miles or 
more, depending on the grades, on the length of the whole 
line, and on the distance between points favorable for the loca- 
tion of shops, etc. The average engine-stage on American 
roads is not far from 75 miles. If there are to be several 
engine-stages on the proposed line, the problem of maximum 
economy of grade must be solved with reference to each of 
| them separately. 

1 | Let Z = length of engine-stage in miles, 

““ @ = expense per engine-mile in dollars, 

‘““ A= average annual paying freight in tons moving in 

one direction, and 

a = average annual paying freight in tons, moving in 
the vupposite direction; and if these are not equal, let A be 
greater than a Now TZ’ eq. (8) is the maximum train-load 
which, at a velocity V, should be hauled up steepest grade 2’ 


tT ge A 
opposed to the direction of the tonnage A; hence 7 = the 


MAXIMUM ECONOMY IN GRADES, ETC. dl 


number of trains per annum; and since each train must go 
2LA f ae , 

and return, .’. —,,;~- = the total train-mileage per annum. 

If there were no return tonnage, the annual expense charge- 

2A Le : 

able to A would be pr but since some of the cars return 
loaded with the freight a, these are not chargeable to A, and 
must be deducted from the above expression. Hence if we 
denote the annual expense of engine-mileage by 2, i 


ie (2A — a) Le 


ae (11) 


in which the value of the maximum grade 2’ is involved in 
the value of 7”. 

But we may obtain an expression for 2 in terms of 2’; for, | 
at any given velocity, the resistance, ¢,, on a level is equal to a | 
the resistance due to a certain grade z,, the value of which is, ay 
by eq. (2), for uniform motion, ay 

39 
o = 74 to 


2 


So the resistance, g, to motion up a grade 2’ is equal to 
9 We 


€ 8 
= g, the total resistance ie 


being that due to the combined grades z+ 2’. Now, since 
the gross weight of a maximum train, under a constant engine 
power, is inversely as the resistances, we have, for conditions 
of accelerated motion: 


the resistance due to some grade z = 


a 


w D 
ri TE ees, ik Ti + Hs: 82, : g2+2 


whence 


ion, ee enna eerste (2) 


in which 7”, = maximum train-load on a level line. Substi- ii 
tuting this value of 7”’ in eq. (11) we have | 
Wit 
| 
By a 
= = an — (2A —a) Le (18) 
37", 0,-—H@ +4 (@—2)) 


t 
w 


which is the complete expression for v = f (z2’) required. 


32 FIELD ENGINEERING. 


66. Could we also find a complete expression for y = f' (2"), 
we might then proceed to find, by analysis, that value of 
e’ which would render #-+-y =a minimum. But the value 
of y cannot be formulated, since it depends on the accidental 
features of the country through which the line passes; it can 
only be determined for any given value of 2 by an estimate 
based on the survey. We therefore resort to a graphical 
solution. 

Equation (13) is the equation of a curve in the plane ZX, 
Fig. 2. If we assume several values—of 2’, and calculate the 
corresponding values of 7, we may lay these off by scale on 
the axes of Z and X respectively, and so obtain several points 


ia. 2. 


through which the curve of annual expense may be drawn. 
We then make estimates of the cost of constructing the road 
at the same values of 2’, and taking the annual interest of 
each estimate as an ordinate y to OZ in the plane ZY, we lay 
it off to scale at the proper height, thus obtaining a series of 
points in the plane ZY, through which the curve of annual 
interest on first cost may be drawn. If now we suppose the 
plane ZY to be revolved to the left about the axis OZ until 
it coincides with the plane OX, as in Fig. 2, we shall see 
that the two curves are convex:to OZ and to each other. The 
shortest horizontal line intercepted by them indicates the 
minimum value of («+ y), and the point where this line cuts 
the axis OZ indicates the corresponding value of 2’, which is 
the one required. If tangents be drawn to the curves at the 
points where the shortest horizontal line intersects them, the 
tangents will be parallel to each other. Any convenient scales 
may be used to lay off the values of z' and 2, provided that 
the values of wand y be laid off to the same scale. It is well 


MAXIMUM ECONOMY IN GRADES, ETC. Ps) 


to reduce all the values of 2 by an amount common to them 
all, and the same with respect to values of y, before laying 
them off to scale. This will bring the two curves nearer 
together without altering their form. 

6%. To facilitate the calculation of 2, we give on the next 

4 1 . , 

page a table of values of via for several engines, using eq. (8) 
for this purpose. The value of vis therefore found, eq. (11) 
or (13), by multiplying (24 — a) Le by the proper tabular 
number, under conditions assumed as follows: 


t- = 10 tons of freight per car-load; 
aw = 18 tons per car and load; 


V = 12 miles per hour. 

Fora 4driver engine, # = 42 tons; P= 8100 Ibs. 

Fora 6-driver engine, #= 49.5 ‘“* P= 12600 “ 

For an 8-driver engine, # = 59.4 ‘“* P= 17280 * 

Substituting these values in eq. (8), and making gq” = 0, 
we find the maximum loads of freight which the several 
engines can haul up the grade whose resistance is g'. The 
reciprocals of these loads are given in the table opposite the 
grades noted in the first and last columns. 

G8. Since qg’ is made zero, the grades in the table are 
assumed to be on straight lines. In locating a road, the 
maximum grade should be reduced on a curve by the amount 
of the equivalent-grade of the curve, eq. (4), so that the resist- 
ance may be no greater on the curve than elsewhere. But 
grades less than the maximum need not necessarily be reduced 
for the curves upon them, unless the sum of the grade and the 
curve-equivalent exceeds the maximum. 

69. For an example, let us suppose that a certain engine- 
stage is to be 80 miles long, and that an estimate of the cost of 
construction has been made, based on a ruling or maximum 
grade of 52.8 ft. per mile against the heavier traffic, and that 
the annual interest on the estimate amounts to $168,000. 

Let us further suppose that the average traffic in one direc- 
tion, is estimated at 375 000 tons per annum, and in the other 
direction at 125 000 tons, that it is decided to use 6-driver 
engines, and that the expense per engine-mile is estimated at 
40 cents; hence (2A — a) Le = 20 000 000. Weare now required 
to find the most economical maximum grade. 

Wefirst select at least two other maximum grades, and having 


FIELD ENGINEERING. 


12 TABLE OF RECIPROCALS OF T’’, b= 104038, 

| | | 

E=2 E=49.5 |-, Fe 280 sae on a ne 

G,. P = 8100 Diff. | P= 12600 | Diff. P = 1780 Diff. ft. per 
| | | mile. 

"9 842 2 5 D RAW | 
BAD His OF79 844: | oy anit) CORAL BBS ees Vint ICR Ror berets 
8.9 | .0457399 | Si a¢9 || -0232 431 | 8 goo || -0157 250 | Pia, | 205.92 
3.8 | .0436.036 | 59 gay |) 0223739 | Q gig || -0151 786 5 336 | 200.64 
8.7 | .0415 679 | 49 yoo || -0215 297 | Bong |} -0146.450 | 2 or9 | 195.36 
8.6 | .0396 259 | 38 x4p || -0207 094 | On4 || .0141 238 | Boos | 190.08 
8.5 | .0877 712 | 37295 || .0199120 | n/5 || .0186 146 | Fong | 184.80 
8.4 | .0359 980 | 46 ogg || -0191 867 | + 545 |] -0181 168 | fone | 179.52 
8.3 | .0343 012 | y¢ ox || -0183 824 | 94) || -0126 802 | Zen» | 174.24 
8.2 | 0826 759 | 35 Fen || -0176 483 |» o4~ || 0121545 | | peg | 168.96 
3.1 |- 10311 176 | 15 583'/| ‘o169 336 | “147 '| 0116 s92 | 48 | 463/68 

Bh eta 14 952 || ae. 6 960 | cial 4553 | - 

.0296 | || .0162 876 | amo || .0112 339 nx | 158.40 
2.9 | ‘028i 864 | 33 So || -o155596 | 6180 1| ‘o1ov sea | 4955 | 153.12 
2.8 | .0268 061 | 43 oem || -0148 988 | ¢ 445 || -0103524 | A ogg | 147.84 
2.7 | 0254 784 | 35 mg || -0142546 | g og5 || -0099 255 | 1189 | 142.56 
2.6 | 0242005 | 35 319 || -0186 264 | g jog || -0095 075 | 4 og4 | 187.28 
2.5 | .0229 695 | 47 gee || -0180186 | 5 oxo || -0090981 | 741; | 182.00 
2.4 | .0217 828 | 44 gar || -0124 157 | 2 age || 0086970 | 3 o39 | 126.72 
2.3 | .0206 881 | 44 o4g || -0118 321 | 5 699 -0083 040 | 3 pr5 | 121.44 
2.2} .0195 333 | 49 gro || -0112 622 |S 566 -0079 188 | gre | 116.16 
2.1 | .0184 663 | ‘ 0107 056 | 2 °P) || .CO%5 413 ‘| 110.88 

2.0 | .0174 352 Bee 0101 6 ome C071 712 tie 
.0174 352 a 20 | C071 712 | on | 105.60 
1.9 | :0164382 | 9 2%2 || ‘0096308 | 2312 || 0068 02 | 3880 | 100.32 
1.8} .0154 736 | gaa || .0091115 | Fong || -0064522 | 3493 | 95.04 
1.7 | .0145 399 | 6 pys || -0086039 | Toe. || .0061 029 | 3 jo» | 89.7 
1.6 | .0136356 | greg || .0081074 7 Qn8 || .0057 602 3 363 | 84.48 
1.5 | .0127593 | 3 4oq || -0076 218 4751 || -0054239 | 339, | 79-20 
1.4 | .0119 099 939 || -0071 467 | 4 gig || -0050 988 | 3 oy49 |+ 13.92 
1.3} .0110860 | »oox || .0066 818 | gre) || .0047 698 | 37g, | 68.64 
1.2} .0102865 | srey || .0062267 | 4 gee || .0044517 | 5 yo, | 63.36 
1.1; .0095104 | ‘‘™ || .0057 810 YF | 0041 393 | | 58.08 
1.0 | .0087568 | yon, || .0053445 | yor, {| .003882 | ao | 52.8 
.0 | .0087 a || .0053 445 Es , Lee | 52.80 
9 | oogozde | F824) ‘0049 101 4277 )| 10085 308 | 20l8") 47.52 
8 | .0073 123 | 6 923 | .0044 984 4104 |, 2082847 | Soi9 | 42-24 
.7 | 0066 200 | gigs || -0040880 | 499; -0029437 Speq 36.96 
-6 | .0059 466 | grx3 || .0086 858 | 3045 || .0026577 Sei, | 31.68 
-5 | .0052.913 | @ 3a || -0082915 | 3 ga || .0023 766 Orgy | 26.40 
4) .0046 583 | & 549 || -0029 050 3791 || -0021002 | gaia | 21.12 
-3 | .0040 320 | @ Ox5 || .0025 259. | Seiq || .0018 284 | 5 Gap | 15.84 
-2| .0034 268 | £ gog || -0021 540 | 3 ay4g | (0015 612 | 969g | 10.56 
-1 | 0028870 | Bren || .0017 892 | 3560 || 0012984 | 9 pen 5.28 
0.0} .0022620 | ° .0014 312 | .0010 399 0.00 


In this table T’ = tons of freight for a maximum train of 
fully loaded cars hauled up any grade ¢z' at a velocity of 12 
miles per hour ; the ratio of dead to paying load being assumed 
at 8to 10. Hence gross load of train behind engine = +5 T’. 
The track is assumed straight, hence q’ = Qin eq. (8) for 


this table. 


MAXIMUM ECONOMY IN GRADES, ETC. 35 
made an estimate of the cost of constructing the road upon 
each, take the annual interest of each, as in the first case. Let 
us suppose the two ruling grades thus selected to be 73.92 ft. 
and 31.68 ft. per mile, or 1.4 ft. per station and 0.6 ft. per 
station, and the interest on the estimates to be $145 596 and 
$204 388 respectively, giving the following statement: 


G;. Y. 1st diff. 2d diff. 
1.4 145 596 29 404 
1.0 168 000 a 13 984 
0.6 204 388 36 35¢ 


Interpolating by second differences, we have the complete 
statement: 


Gar) Ye. | diff. y. | Cre. | gs ety. z 

1.4 | 145596 ohh ee | 142 934 eatery a2 

1.3 | 149886 Sr ietisron & | 2 Gs ae | 

1.2 | 155 050 6038 | solid 124 534. | 

1.1 161 088 6912 | 8730 115 620 | | 

1.0 168000 | wrge | gig | 106 890 74 890 52.80 

0.9 175786 | ges | gag | 98342 | 274128 47.52 

0.8 184 446 9534. | g0g | 89968 | 274414 42.24 

0.7 193 980 1408.1 e044. 1, SL OO 

0.6 | 204388 | | | @u6 | 31.68 
| j 


The numbers in the fourth and fifth columns are obtained 
as follows: the values assumed above give us (2A — a) Le = 
$20 000 000, and this multiplied by the tabular differences in 
the preceding table for a 6-driver engine, gives the numbers in 
the fourth column. We now observe that the differences of 
z and of y increase in opposite directions, therefore at some 
point they will be equal; and a simple inspection shows us that 
this point is at or near the grade of 0.9, which is therefore 
the grade required. We now multiply the tabular number for 
0.9, and a 6-driver engine by $20 000 000, for the number in 
the fifth column, and this added to the value of y on the 
same line gives the sum of («+ y) for the most economical 
grade. This of course is not the total annual outlay of the 
road, or engine-stage, because many items of expense which 
are independent of a maximum grade have not been con- 
sidered. 


36 FIELD ENGINEERING. 


If an 8-driver engine were to be used, and the expense per 
engine-mile estimated at 50 cts., then (2A — a) Le = $25 000000; 
hence 


G,. y. diff. y. diff. a. i woty. z'. 
ita! 161 088 ae 

6 912 % 6700 4 a 
ce be oat 7 286 538 95 810 263 810 52.80 


indicating a saving of $10 818 per annum in the case supposed 
by using 8-driver engines, although on a steeper ruling grade. 
On the other hand, should we adopt 4-driver engines, and esti- 
mate the expense per engine-mile at 30 cents, we should find 
the most economical grade to be 0.7 per station and (# + y) 
= $2938 280, showing a loss in this case of $19 152 per annum, 
as compared with the results of 6-driver engines. 

It should be remembered that the table § 67 is prepared on 


: ROT, eordl 
the assumption that the ratio ea es If cars are to be used 
giving for full loads any other ratio,. 5 a new table may be 
LO = snag 
prepared by multiplying each tabular number by 78 xs = 


The velocity adopted of 12 miles per hour is sufficient a 
ordinary grades. When the maximum grade is very low, it 
would be better to use 15 or 18 miles an hour in calculating 
the value of 2. 

70. Since z, eq. (11), varies directly as Z, it is important 
that an engine-stage having heavy grades should be short. Its 
length, however, must be consistent with the economical 
length of the adjoining engine-stages, and with the amount of 
work which an engine ought to perform daily. The most 
favorable condition for a road would be that in which all the 


_ engine-stages were operated at equal expense. But if, to 


secure this result, the engine-stage of heavy grades must be 
unreasonably reduced in length, it will be better to adapt the 
grades to the use of two engines per train. 

71. The maximum grade 2’, opposed to the heavier tonnage 
A, having been determined, we have now to consider what is 
the limit to grades in the opposite direction. The engines are 


MAXIMUM ECONOMY IN GRADES, ETC. 37 


supposed to haul their maximum loads in moving the ton- 
nage A, and since the return tonnage, a, is less than <A, the 
engines, in returning, will not be worked to their full capacity 
if they encounter no grades steeper than ¢’. We therefore 
have a margin of power in the returning engines which may 
be taken advantage of to cheapen the cost of construction, or 
to shorten the line, by introducing grades, steeper than 2’, 
against the lighter traftic. | . 
The weight of a maximum train moving up the grade ¢’ is, an a 
eq. (9), W'-+ 7”; the weight of the train returning will be Wh 


¢ 


al Diy w—t a ! 
pen el Tits (a BE 


Substituting this in place of (W’‘ + 7’), eq. (9), and solving for Wi 
q, we find the resistance due tu a maximum grade opposed to GE 
the returning train. Whence, by eq. (2), if we let Z=the A) 
maximum return grade, and make q” = 0, 


Se pea au 


w—t , 24 Wa 


Inasmuch as the value of Z varies with every change made 
in 2’, the engineer, when estimating the cost of construction 
upon the basis of any maximum grade 2’, should take care 
that the return grade Z nowhere exceeds its limit as given by 
the last equation (14). In the example, $69, 2’ = 47.52; hence 
T’ = 208.37, eq. (8). Substituting these values, in eq. (14), we 
find Z = 81.25, which is therefore the limit for return grades 
in this case. With regard to curves on the maximum grade, 
see $68. 


72. If ineg. (1) weletz = 


33 
14 
offers a resistance equal to the resistance to uniform motion 


on a level, we have 


q be the grade per mile which 


00141422 | 
= To Son ee 7% 15 | 
2= 12.784 (01414 + pe) (15) | 
When V = 20 this becomes | 
~ 2 Wy 
BES 8 BOE); git en AE Me! (153) 


E+W+T 


3 FIELD ENGINEERING. 


which is the grade down which a train, whose weight is (H 
W-+ T), if started at 20 miles an hour, will continue to move 
at that speed without steam or brakes. As that speed is not 
objectionable, so the grade 2, which induces it is not, pro- 
vided it does not exceed the values of 2’ or Z respectively, 
determined with reference to economy. For the extra work 
done by the engine in ascending one grade ¢ is utilized in 
descending the next; and the net result is the same as though 
the two were replaced by a uniform grade. The engineer 
therefore is not warranted by economic considerations in 
reducing undulating grades which do not exceed z to a uni- 
form grade, when to do this would cause any increase in the 
cost of construction, unless 2 exceeds the grades z' or Z of 
maximum economy. . 

73. But when grades exceed 2, eq. (154), the resulting 
speeds of the maximum train become too great, and the neces- 
sary application of the brakes absorbs a portion of the power 
previously expended in gaining the summit, which is thus 
worse than wasted, since it increases the wear and tear of 
machinery and track. Therefore the engineer is justified in 
spending a certain sum of money in reducing grades which 
exceed z to that limit. A calculation of the loss of power due 
to the use of brakes on a grade, and of the cost of that lost 
power, together with the resulting wear and tear per annum, 
will give the interest on the sum that may be justifiably spent 
in reducing the grade from its position of cheapest construc- 
tion. 

74. The limit 2 is not constant, but depends on the weight 
of the maximum train, which in turn depends on 2’. It will 
not be the same in both directions unless A =a, giving 2’ = Z. 
In the example §69, # = 49.5 and W' + 7’ = 366.07; hence, 
eq. (155), 2 = 21.72 descending in the direction of the traffic A. 


Also W' +4 T' = 230.49, whence z = 23.34 descending in 


the opposite direction. These are the limits in this case at 
which undulating grades cease to be profitable. 

75. We have finally to consider the method for selecting the 
best line from several proposed routes. For this purpose we 
determine the most economical grade on each route thought 
worthy of consideration, and calculate the interest on the 
entire cost of constructing the line with that ruling grade, and 


S 


LOCATION. 39 


also the annual expense of operating the line, and take the sum 
of the two. ©That route is best in respect to which this swm is 
the least. 

7G. The value of saving onc mile in distance on any route 
is found by dividing the sum of the annual operating expense 
and the interest on the cost of construction by the rate of 
interest, and the quotient by the length of the line in miles. 

77. We have now fully discussed the theory and developed 
the formule necessary to the determination of the most i] 
economical grades; but the value of the results in a given i 
case depend upon the correctness of the engineer’s estimates 
which enter into the formule. These may seldom prove pre- . 
cisely accurate, yet, if he can bring them within definite Wy 
limits, he may determine the grades of maximum economy 
within corresponding limits. In the case of a finished road 
and in full operation, however, the elements of first cost, of Hh | 
traffic, and of operating expenses being known, an investiga- 
tion by means of the foregoing formule becomes a critical test 
as to the economy of the location and grades; and should the 
road fail to pay dividends, or be forced to charge high rates 1 
of toll, we can determine, though perhaps too late, to what | 
extent the location is chargeable with these results. ii 


CHAPTER IY. ay 
LOCATION. 


78. A railroad is said to be located when its centre line is 
established on the ground in the position which it is intended 
finally to occupy. The location is made by an engineer corps | 
similar in its organization to that employed on preliminary | | 
surveys. The instruments used are also the same, except that VW 
the transit is substituted for the compass, and usually the target | | 
rod’ for the self-reading rod. The magnetic needle is never Hi 
used upon the centre line, except as a rough check on the 
transit work. It is used, however, to obtain the direction of 
property lines, roads, and other topographical data. 

79. The remarks upon transit work in the preceding 
chapter apply to the running of straight lines on location. All 1 


40 FIELD ENGINEERING. 


field-work on location should be done with accuracy and 
fidelity. No guesswork, nor rude approximations, are to be 
tolerated. AJ] transit points are made as secure and permanent 
as possible, and the more important ones are guarded by other 
transit points set in safe positions near by, their distances and 
directions from the main point being recorded. 

The stakes for the stations are made ne atly, and somewhat 
uniform in size, and they are firmly driven. Sometimes a 
small plug is driven down flush with the surface of the ground 
to indicate the station point, and the stake is then set near by 
as a witness. 

In locating a very long tangent the greatest care is re- 
quired to make it straight. If the tangent is produced from 
point to point by backsights and foresights, the observation 
should be repeated in every instance with reversed instrument, 
to eliminate any possible lack of adjustment, and to check 
any accidental error. (Indeed it is proper to observe this rule 
on curves, as well as on tangents.) When some object in the 
horizon can be used as a foresight, it is preferable to set the 
instrument by this rather than by a backsight. For final loca- 
tion, the line should be cleared to give as continuous a line of 
sight as possible, but in case of an obstacle which cannot be 
removed at the time, at least two independent methods of 
passing it should be employed, so that there may be a check 
upon the alignment beyond. 

SO. The leveller selects his benches far enough from the 
line to prevent their being disturbed during the construction of 
the road. They should be nearly at grade, as a rule, though it 
is well to leave a bench near a water-course for reference in lay- 
ing out masonry of trestle-work. The rodman holds the rod 
at every station, and at every point on the centre line where 
the slope changes direction, so that these points may be accu- 
rately defined on the profile. When he uses a target rod, he 
sets the target as directed by the leveller, and after clamping 


‘it, takes the reading. He reads to thousandths upon turning 


points and benches, but only to tenths of a foot elsewhere, and 
announces the readings to the leveller for record. He also 
records the readings upon turning points and benches in his 
own book asa check. At the close of each day the leveller 
and rodman compare notes, and draw a profile of the line sur- 
veyed. (See also §§ 28, 29, 30.) 


LOCATION, 


$1. The fixing of the grade-lines upon the profile is 
one of the most important operations connected with the loca- 
tion. It is usually performed by the engineer in charge of 
the locating party, as being most conversant with the general 
character and detailed requirements of the line. The maxi- 
mum gradients will have generally been determined in advance 
from the preliminary data by the principles laid down in the 
preceding chapter, but the position of each grade-line, relative 
to the profile of the surface, must be left to the judgment and 
skill of the engineer. In general, the grade-line is so placed 
as to equalize the amounts of excavation and embankment, 
but there are various exceptions to this rule. Thus, the exca- 
vation may be in excess: first, when it is necessary to pass 
under some other road or highway, the grade of which cannot 
be changed; second, when valuable property is to be avoided, 
the appropriation of which would cost more than the excava- 
tion; third, when the grade is at the maximum near a sum- 
mit, and cannot be raised parallel to itself without incurring 
too great an expense for masonry, etc., at some other part of 
the line. The embankment may be in excess, first, when the 
country is flat and wet, in order to keep the road-bed well 
drained; (the grade-line should be at least two feet above tke 
average level of the surface, or above high-water mark, if the 
district is subject to overflow;) second, in approaching a 
stream, where it is necessary to raise the grade above the 
requirements of navigation; third, when the cuttings on the 
line are largely in solid rock, and a cheaper material for 
embankments may be conveniently had at other points; 
fourth, in a district subject to heavy drifts of snow, by which 
deep cuts would be liable to be obstructed; fifth, in side-hill 
work, where there is danger of land-slips; sizth, when it is 
determined to supply the place of a portion of an embankment 
by a timber trestle-work or other viaduct. 

The apparent equality of cut and fill on the profile does not 
represent an equality in fact, owing to the different bases and 
slopes of the sections adopted, and to the various inclinations 
of the natural surface transversely to the line. This is espe- 
cially true in side-hill work, where there are both cut and fill 
at every point, while the profile shows very little of either. In 
the latter case it is an excellent plan to combine with the pro- 
file of the centre line the profiles of parallel lines ten 


4° FIELD. ENGINEERING. 


or twenty feet either side of the centre, and drawn with differ. 
ent colored inks, as these will indicate tolerably well the relative 
amount of cut and fill required. But after the grade has been 
thus chosen, the only safe method in side-hill work is to 
actually compute the amounts of excavation and embankment 
from cross-sections, mark the amount for each cut and fill on 
the profile, and compare the results. Any changes required in 
the grade or alignment may then be discovered and effected 
before the work of construction has begun. 


C. H-A.P TER. .Ve 
SIMPLE CURVES. 
A. Hlementary Relations, 


$2. The centre line of a located road is composed alternately 
of straight lines and curves. 

The straight lines are called tangents because they are laid 
exactly tangent to the curves. <A tangent may be indefinitely 
long, but should never, as a rule, be shorter than 200 feet 
between two curves which deflect in opposite directions, nor 
shorter than 500 feet between curves which deflect in the same 
direction. A curve should not be less than 200 feet long. 
When a tangent is said to be straight, the meaning simply is 
that it has no deflections to the right or left; for since it fol- 
lows the surface of the ground, it evidently has as many 
undulations as the ground. But if we conceive a vertical 
plane to be passed through the line, a horizontal trace of this 
plane will accurately represent the line; and so, if we con- 
ceive a vertical cylinder to be: passed through a curve on the 
surface of the ground, a horizontal trace of that cylinder will 
accurately represent the curve, since all distances and angles 
are measured: horizontally, whatever be the irregularities of 
the surface. In all problems, therefore, relating to this sub- 
ject, we may consider the ground to be an absolutely level 
plain. 

83. A Simple curve isa circular arc joining two tan- 
gents. It is always considered as limited by the two tangent 


SIMPLE CURVES. 


points, and any part of it beyond these points is called the 
curve produced. The first tangent point, or the point where 
the curve begins, is called the Point of Curve, and is indicated 
by the initials P.C. The point where the curve ends, and the 
next tangent begins, is called the Point of Tangent, and is indi- 
cated by the initials P.7. When accessible, these points are 
always occupied by the transit in the course of the survey, 
and the plug driven to fix the point is guarded, not only by 
the usual stake bearing the number of the station, but also by 
another bearing the proper initials, the ‘‘ degree” of the curve, 
and an ‘‘R” or ‘‘L” to indicate whether the deflection is to 
the Right or Left. 

S4. A simple curve is designated either by the radius, R, 
or the degree of curve, D. 

The Degree of Curve, D, is an angle at the centre, sub- 
tended by a chord of 100 feet. It is expressed by the number 
of degrees and minutes in that angle, or in the arc of the 

curve limited by the chord of 100 
b feet. Therefore D equals the num- 
ber of degrees of are per station. 
Q The radius R and degree of 
, curve J) can be expressed in terms 
of each other. 
Let ab, Fig. 3, be a chord of 

100 feet subtending an arc de- 
scribed with a radius a0 = Rk from the centre 0. Then, by 
definition the angle bea = D. Biscct the angle boa by a line 
og, and this line will also bisect the chord ad and be perpen- 
dicular to it; and in the right-angled triangle dgo we have 


Fie. 3. 


bg = ob X sin bog 
or 


a = sin 4D 


Hence, to find Radius in terms of Degree of Curve: 


50 
dea a. 16 
t sin 4D 0 


and to find Degree of Curve in terms of Radius: 


She seers (177) 


I 


& 


44 FIELD ENGINEERING. 


It is the practice of English engineers to assume the radius 
at some round number of feet and calculate the degree of curve, 
which is therefore fractional. In America, on the contrary, 
the degree of curve is assumed at some integral number of 
degrees or minutes, and the radius deduced from this. 

Heample.—W hat is the radius of a 3° 20' curve? 


50 log 1.698970 
4D = 1° 40’ log sin 8.463665 


Ans. R=1719.12 log 3.235305 
Thus the second and third columns of Table IV. have beew 
calculated. 


Example.—W hat is the degree of curve when the radius is 
600 feet? 


50 log 1.698970 
R=-600 log 2.778151 


4D = 4° 46' 48".'73 log sin 8.920819 
Ans, D = 9° 33' 37" .46 


Measurement of Curves. 


85. A railroad curve is always assumed to be measured with 
a 100-foot chain, and as the chain is stretched straight between 
stations it cannot coincide with the arc of the curve, but 
forms a chord to the arc, as in Fig. 3. Consequently the 
curve as measured from one tangent point to the other is an 
inscribed polygon of equal sides, each side being 100 feet. 
The sum of these sides (with any fraction of a side at either 
end of the curve) is called the Length of curve, L. This length 
L is evidently a little less than the length of the actual are 
between the same points, but the latter we very seldom have 
occasion to consider. 

8G. If the chain lengths were taken on the are instead of as 
chords of the curve, the degree of curve would be inversely 
proportional to the radius, and since the are whose length is 
equal to radius contains 57.3 degrees nearly, we should have 


DS OTF e100 © ie. 
or 
_ 5180 


PD 


SIMPLE CURVES. 45 
a convenient formula, but only approximately true when D is 
small, and seriously at fault when D is large; the error in- 
volved being proportional to the difference in length of a 
100-foot chord, and the are which it subtends. 

87. The Central Angle of a simple curve is the angle 
at the centre included between the radii which pass through the 
tangent points (P.C.) and (P.7.). It is therefore equal to the 
number of degrees contained in the entire arc of the curve 
between those points. The central angle will be designated 
by the Greek letter A (delta). 

From the definitions of the length and degree of curve we 
have the proportion, 

LTE. = 100371. 


fence, to find the Length of curve én terms of the central 
angle: 


A 
23 18 
EL = 100 i ( ) 


Example.—W hat is the length of a 4° curve when the cen- 
tral angle is 29°? 
D =A’ and A= 20° § 4)2900 
Ans. I= stations + 25 feet ( 725 feet. 
To find the Central angle in terms of the length and degree 
of curve: 


Livample.—What is the central angle of a 5° curve 730 feet 
long? 
. 
Io: cea TOU; ca WV res, 
Ans. A = 36° 30' 


To find the Degree of curve in terms of the length and 
central angle: 
— 400 2 20) 
D = 100 = ( 
Example—What is the degree of a curve 8 stations long, 
and having a central angle of 26° 40'? 
26°.666 


— = 3°.333 


L = 800, A = 26°.666, 100 800 


Ans. D = 8° 20’ 


46 FIELD ENGINEERING. 


88. If two tangents, joined by a simple curve, are produced 
(one forward and the other backward) until they intersect, the 
point of intersection, V (Fig. 4), 
is called the vertex, and the 
exterior or deflection angle 
which they make with each 
other is equal to the central 
angle, A 

The Tangent-distance, 
T, is the distance from the 
vertex to either tangent point; 
thus in Fig. 4, 7= AV=VB. 

The Long Chord, C, is 
the line APB joining the two 
tangent points. 

Fra. 4. The Middle-ordinate, 

M, is the line GH, joining the 

middle point of the long chord with the middle point of the 
curve. 

The External distance, #, is the line HV, joining the 
middle point of the curve with the vertex. 

We observe that both the middle-ordinate, J, and the 
external distance, #, are on the radial line joining the centre, 
O, with the vertex, V, and that this line is perpendicular to 
the long chord, C; also, that it bisects the central angle 
AOB= A, and its supplement AVB. (Tab. 1.14.) We also 
observe that the angle VAB= VBA =A (Tab. I. 20); and 
if in the figure we draw the two chords AH and BH, the 
angle BAH equals one half the angle BOH, or BAH = ABH = 
4A (Tab. I. 18); also the angle VAH= VBH=4a. 

89. If we have laid out two tangents on the ground, inter- 
secting at V, and have measured the angle, A, between them, 
we may then assume any other one of the elements of a 
simple curve before mentioned, and calculate the rest. If 
we assume JD, for instance, we then find & by eq. (16) or by 
Table LY. 

Then, having A and R,we may proceed to. calculate the 
other elements as they are needed. 

90. To find the Tangent-distance in terms of the 
Radius and Central Angle: 


4 | 


SIMPLE CURVES. 


In the right-angled triangle VOA, Fig. 4, we have 


VA = OA X tan VOA 
Pl oh ta FA (21) 


Otherwise, approwimately: In Table VI., opposite the central 
angle, take the value of 7 for a 1° curve and divide it by the 
degree of curve D. If desirable, add the correction taken 
from Table V., corresponding to D. 

Hxample.—What is the tangent distance of a 4° curve with 
a central angle of 30°? 


Diese R(Table IV.) log 3.156151 
A= BO, 4A = 15° _ log tan 9.428052 


Ans. T' = 3838.89 feet log 2.584208 
Otherwise: 
By Table VI. 4)1585.3. 
Approximate ans. 383 . 82 
Correction from Table V. .08 
Ane Tt = 383.90 feet. 


91. To find the Long Chord (C, in terms of Radius and 
Central Angle: 
In the right-angled triangle BOG, Fig. 4, we have 


BG = BO X sin BOG 
or 
4C0=Rsinta 
.C=2Rsinta (22) 


But in case A can be divided by D without a remainder, 
that is, if the curve contains an exact number of stations (not 
exceeding 12), we may take the long chord at once from 
Table VII. 

Example.—What is the long chord of a 3° 20’ curve with a 


central angle of 86° 40' ? 


2 log 0.801030 
=o 20. (lab. LV.) log 3. 2353805 
A = 36° 40’, 4A = 18° 20' log sin 9.497682 


Ans. C = 1081.48 feet log 3.034017 


A8 FIELD ENGINEERING. 


Otherwise: 
A 863° 
ie —-- = |{ stations 
pr BP 


And by Table VII. O = 1081.48. 

92. To find the Middle-ordinate Y, in terms of Radius 
and Central Angle: — 

It is evident from the figure that if the radius OH were 
unity, the line GH would be the nat. versed sine of the arc 
BH. But the arc BH measures the angle BOH= 4A, and 
Ofek: 


. M= Rversta (23) 


But in case A can be divided by D without a remainder, 
. that is, if the curve contains an exact number of stations (not 
i exceeding 12), we may take the middle-ordinate at once from 
Table VIII. 

Example.—What is the middle-ordinate of a 4° 30’ curve 
with a central angle of 40° 30’? 


| Dawa R (Tab. IV.) log 3.105022 
| A = 40° 30, 4A = 20° 15’ log vers 8.791049 


Ans, Ma U8 7 1.896071 
Otherwise: 
A> 4075 
9 Aldo 9 stations 


and by Tab. VIII. M = 78.717 
93. To find the External Distance £# in terms of 
Radius and Central Angle. 
It is evident from the figure that if the radius OA were 
i unity, the portion HV of the secant line OV would be the 
i external secant of the arc AH. But the arc AH measures the 
1 || angle AOH= 4A, and OA=Rf; 


‘ H= Rex see ga (24) 


Otherwise, approximately: 

In Table VI., opposite the central angle, take the value of 
E for a 1° curve, and divide it by the degree of curve D. 
If desirable, add the proper correction corresponding to D, 
taken from Table V 


SIMPLE CURVES. 


Hrample.—W hat is the external distance # of a 7° 30’ curve 
when the central angle is 60° ? 


DT tau, ?(Tab. IV.) log 2.883371 
LOU", 4A = 80° log ex sec 9.189492 
Ans. H = 118.27 feet log 2.072863 
. Otherwise: 
By Tab. VI. 7.5)886.38 
: Approximate ans. 118.184 
Correction for D = 7° 30' (Tab.. V.) 084 
Ans. H= 118. 268 


94. But, instead of assuming D or R, we may prefer, or may 
find it necessary to assume, some other element of the curve, 
the central angle being given. 

If we assume the tangent distance, then: 

95. To find the Radius and Degree of Curve in terms 
of the Tangent-distance and Central Angle. 

From eq. (21), and by Table II. 40, we have 


R=T cotia (29) 


Otherwise, approximately: 

Divide the tangent of a 1° curve found opposite the value of 
A in Table VI., by the assumed tangent distance; the 
quotient will be the degree of curve in degrees and decimals. 

Kxample,—The exterior angle at the vertex is 54°, and the 
tangent distance must be about 700 feet. What shall be the 
degree of curve? 


A = 54°, 4A = 27° log cot 0.292834 
Rig. 5 (X) 2.845098 
log B= 3.137932 
Ans. By Table IV. D= 4° 10’ + 


Otherwise: 


pe 


By Table VI. 700)2919.4 
Ans, D = 4° 10' 15” 4.170 


=r) 


But as it is difficult to lay out a curve when JD is fractional, 
we discard the fraction and assume 4° 10’ as the value of D. 


50 FIELD ENGINEERING. 


This may require us to recalculate the value of 7, which we 
do by eq. (21) and find 7’= 700.8 feet log 2.845596. If the 
other elements are required, they may be calculated by eqs. 
(22), (23), (24), or directly from 7’ and A, as follows: 

96. To find the External distance LH, in terms of the 
Tangent-distance and Central Angle. 

In Fig. 5 we have given 

AOB = A. and.AV = 7 stone 
HV=&. In the diagram draw 
the chord AH, and through H draw 
a tangent line to intersect OA pro- 
duced in J, and join VZ. 

Then HI is parallel to BA, and 
since HT = AV= 7, and Al= HV 
— BH, VI is parallel to HA, and 
Vit = STAB= tet Tababasy 
In the right-angled triangle VHI we have 


Fe. 5. 


HV = HI x tan VIH 
or H= Ttan}fa (26) 


Erample.—The angle at the vertex being 54° and the tan- 
gent-distance 700.80 feet, how far will the curve pass from 
the vertex ? 


T = 700.80 (from last example) 2.845596 
A = 54°, 4A = 138° 30’ log tan 9.880354 


foo] 


Ans. H = 168.25 feet log 2.225950 


(For the formule by which to find the long chord and mid- 
dle-ordinate in terms of the tangent-distance and central angle, 
see Table III. 12 and 13.) 

97. Again, it may be necessary to assume the eaternal dis- 
tance in order to determine the proper degree of curve. 

To find the Radius and Degree of Curve in terms of 
the External distance and Central Angle: 

By eq. (24) 

R238 @) 


ex sec dA 


SIMPLE CURVES. 
Otherwise: 
In Table VI. divide the external distance of a 1° curve, 
opposite the given valuc of A, by the assumed -external dis- 
tance; the quotient is the degree of curve required. 
Hrample.—The angle at the vertex being 24° 30’, the curve is 
desired to pass at about 65 feet from the vertex. What is the 
proper degree of curve ? i 


E=65 log 1.812913 

A = 24°30', 4.4 = 12° 15’ log ex sec 8.367345 | 
eRe 3.445568 | 

ans, By Table IV. D = 2° 03’ + i 


Otherwise: 
_ By Table VI. . 65)133.50 
Ais: D= 2° 03’ 14” 2° .0538 


We may therefore assume a 2° curve, unless required by 
the circumstances to be more exact, when we might use a 
2° 03' curve. Assuming a 2° curve, we have by eq. (24) 


E = 68.75 log 1.824460 


Having decided on the degree of curve, we may calculate 
the remaining elements by eqs. (21), (22), (23), which is always 
the better way, but we may calculate them directly from # 
and A. 

98. To find the Tangent-distance in terms of the 
External distance and Central Angle: 

From eq. (26), and by Table IT. 40, 


T = E cotta (28) | 


t 
| 
Example.—The angle at the vertex is 24° 80’, and the curve i 
passes 66.75 feet from the vertex. How far are the tangent | 
points from the vertex ? 


i = 66.75 (from last example) log 1.824460 


S 


A = 24° 30', $A =6° 0730" _ log cot 0.969358 
Ans. T = 622.04 feet 2.793818 


99. Remark.—Kqs. (27) and (28) are particularly useful in 
defining the curve of a railroad track where all original 


on FIELD ENGINEERING. 


points are lost. Produce the centre lines of the tangents of 
the curve to an intersection V, and there measure the angle A. 
Bisect its supplement AVA, and measure the distance on the 
bisecting line from V to the centre line of the track. This 
will give VH= H. Then 2 and 7 may be calculated, and the 
distance 7’ laid off from V on the tangents, giving the tangent 
points A and B. 

(For the formule by which to find the long chord and mid- 
dle-ordinate in terms of # and A, see Table III. 16 and 17.) 

100. Again, having only the central angle given, we may 
assume the long chord, or the middle-ordinate, and from either 
of these and the central angle calculate the remaining ele- 
ments. Or, finally, the central angle being wnknown, we may 
suppose any two of the linear elements given, and from these 
calculate the rest. As such problems have little practical 
value, their discussion is omitted. The requisite formule for 
their solution are given in Table III., and the verification of 
them is suggested as a profitable exercise to the student. 


Bb. Location of Curves by Deflection Angles. 


101. In order that the stakes at the extremities of the 
100-foot chords, by which the curve is measured, shall be set 
exactly on the arc of the curve 
by transit observation, it is neces- 
sary at the point of curve, A, to 
deflect certain definite angles 
from the tangent. AV. Let us 
suppose that in the curve AB, 
Fig. 6, the points A, a, 0, c, d, 
etc., indicate the proper posi- 
tions of the stakes 100 feet apart, 
and that OA is the radius of the 
curve. In the diagram join Oa, 
Ob, ete.,,and also Aa, ab, be, ete. 
Then, by definition, the angle AOa=D, and by Geom. 
(Tab. I. 20 and 11) the angle VAa = 4D. ‘Therefore if 
we set the transit at A, and deflect from AV the angle 
4D, we shall get the direction of the chord Aa, on which by 
measuring 100 feet from A we fix the stake, a, in its true 
position on the curve. So again, since the angle a@0d, at 
the centre, = D, the angle qAd, at the circumference, = 1D. 


Fie. 6. 


SIMPLE CURVES. 
If therefore, with the transit at A, we deflect the angle 3D 
from the chord Aa, we shall get the direction of the chord 
Ab; and when the stake d is on this chord it will also be on 
the curve, if 6 is 100 feet distant from a. Thus, in general, 
we may fix the position of any stake on the curve, by deflect- 
ing an angle 1D from the preceding stake, and at the same 
time measuring a chain’s length from it,—the chain giving 
the distance, while the instrument at A gives the direction of 
the point. 

1D) is called the Deflection-angle of the curve; so that in 
any curve, the deflection-angle is equal to one half the degree of 
curve. 

102. Since each additional station on the curve requires 
an additional deflection-angle, the proper deflection to be made 
at the tangent point from the tangent to any stake on the 
curve is equal to the deflection-angle of the curve multiplied 
by the number of stations in the curve up to that stake; or it 
is equal to one half the angle at the centre subtended by the 
included arc of the curve. 

103. It may happen that all the stations of a curve are not 
visible from the tangent point, A. When this is the case a 
new transit-point must be prepared at some point on the 
curve, by driving a plug and centre in the usual manner, and 
the transit moved up to it. Let us suppose that the point d, 
Fig. 6, has been selected for a transit-point, and that the 
transit has been set up over it. Before the curve can be run 
any farther, it is necessary to find the direction of a tangent to 
the curve at the point d. Forthis purpose we deflect from 
chord dA an angle Adz equal to the angle V-Ad previously 
deflected to fix the point d. (Tab. 1.16.) Or we may adopt the 
following 

Rule: To find the direction of the tangent to a 
curve at the extremity of a given chord, deflect from the chord an 
angle equal to one half the angle at the centre subtended by the 
chord. (Tab. 1. 20.) 

Having thus found the direction of the auxiliary tangent 
zdz, we proceed to deflect from dz, (3D). for the next station e¢, 
2 (LD) for station f, 3(4D) for station g, etc., as before. When 
the end of the curve is reached, a transit-point is set at the 
Point of Tangent, after which it only remains to find the 
direction of the tangent, by the above rule. Thus if g is to be 


AA. FIELD ENGINEERING. 


the point of tangent, we obtain the direction of the tangent by 
deflecting from the chord gd an angle equal to zdy, or to 
x 40g. If this tangent VB was already established, the line 
ge thus obtained should coincide with it; and if it does so, 
the correctness of our work is proved. 

104. The centre line is measured, and the stations num- 

ered regularly and continuously through tangents and 
curves from the starting point to the end of the work. It 
therefore frequently happens that a curve will neither begin 
nor end at an even station, but at some intermediate point, or 
plus distance. 

If the Point of Curve occurs a certain number of feet 
beyond a station, the first chord on the curve is composed of 
the remaining number of feet required to make 100. 

Any chord less than 100 feet is called a subchord. 

If a curve ends with a subchord, the remainder of the 100 
feet must be laid off on the tangent from the Point of Tangent 
to give the position of the next station, so that the stations 
may everywhere be 100 feet apart. 


105. The deflection to be made for a subchord is equal to one 
half the arc it subtends, 
Let c = length of any subchord in feet. 
“ad = angle at centre subtended by subchord. 
Then, from eq. (22), by analogy 


¢ =2R sin id (29) 
100 
; Oy? pt Meee 
But by eq. (16) EE sinteD 
sin 1d 
6 = 100 (30) 
- sindd ——°~ gin 1D (31) 
i 100 3 


When D does not exceed 8° or 10°, we may assume without 
serious error that the angles are to each other as their sines, 
and the last two equations become 


(approx.) e = 100 (82) 


SI a 


s i 


SIMPLE CURVES. 


and id = —— (iD) (33) 


In curves sharper than 10° per station, the error involved in 
this assumption becomes apparent and must be corrected. 

106. If curves were measured on the actual arc, then 
eqs. (82) and (33) would be true in all cases; but since a curve 
is ‘measured by 100-ft. chords, it is evident that if a 100-ft. Hl) 
chord between any two stations were replaced by two or more | 
subchords, these taken together would be longer than 100 feet, 
since they are not in the same straight line. Let us conceive 
the actual arc of one station to be divided into 100 equal 
parts; since the arc is longer than the chord, each part will be ni 
slightly longer than one foot. Now if we take an arc contain- 
ing any number of these parts (less than 100), the nominal 
length of the corresponding subchord in feet will egual the 
number of parts, and the deflection for the subchord will be 
proportional to the number of parts which the arc contains. Mi il 
The deflection therefore will be exactly given by eq. (83) if in Ali 
that equation we let c equal the number of parts in the arc, or 
the nominal length of the subchord in feet. Having thus 
obtained the correct value of ($d), we may introduce it into 
eq. (29) or (80), and obtain the ¢rwe value of the subchord, 
which will always be a little greater than its nominal value. . 

Suppose, for instance, that the arc of one station is to be wat 
divided into four equal portions; then each subchord will be , 


29 nia 


nominally 25 feet long; and by eq. (33 


95 | Hl 

$d = GD) =2 GD) a 

which is the correct value of the deflection, whatever be the 
degree of curve. Substituting this value in eq. (29) or (380) we i 
obtain the true value of the subchord, ¢, a little greater than AWA 
25; the eacess is called the correction of the nominal length. 1 
107. This correction for any given subchord bears an Hy 
almost constant ratio to the excess of are per station, what- | | 
ever be the degree of curve. These ratios are shown in the WH 
following table for a series of subchords, and Table VII. gives | 
the length of actual are per station for various degrees of 
curve. Subtracting 100 we have the excess of arc per station, 
and multiplying this evcess by the ratio corresponding to the 


56 FIELD ENGINEERING. 


nominal length of subchord we obtain as a product the proper 
correction for the subchord. 


TABLE OF THE RATIOS OF CORRECTIONS OF SUBCHORDS TO 
THE EXCESS OF ARC PER STATION. | 


Nominal | Nominal Nominal ; 

Length of Ratio. || Lengthof| Ratio. || Length of | Ratio 

’ Subchord. | | Subchord. Subchord. | 
—_____! a a { 

0 .000 35 .207 70 .3806 

5 .050 40 .oa5 (45) Ry 

10 1099. .o4f) SU4R That oer papeei> | G0 STs Nad [ogy 

15 .147 50 .Ot4 85 .205 

20: .192 55 | .083 90 .169 

25. .234. 60 .883 95 | .092 

30. eels 65 374 100 .000 


We observe that the largest correction is required by a sub- 
chord between 55 and 60 feet in length. 

Example.—It is proposed to run a 14° curve with a 50-ft. 
chain. What correction must be added to the chain? 


= ig a — 7° By _ 90 Mo 90 & __ g0 arp 
D== 14 iD=% Mie (~ = 3°.5 = 8° 80 
By eq. (30) 
6100 ee nes 
sin 7 


Ans. Correction = .093 


Or, by Table VIL., length of arc = 100.249 


excess of arc = 249 
and by above table, ratio for 50 feet = .Bt4 
Ans. Correction = product = .093 


Example.—The P.C. of an 18° curve is fixed at + 55 feet 
beyond a station. What are the nominal and true values of 
the first subchord, and what the proper deflection? 


Nominal value = 100 — 55 — 45 feet 

; 45 s 
Deflection = id= 100 xX 9° = 4°.05 — 4° 03’ 
and by eq. (80) 
True value = c = 100 oe ee 


she Uda 45.148 


Gq 


SIMPLE 


CURVES. 


Or, by Table VII., excess of are= _ _—-. 412 
by above table, ratio for 45 feet = _.358 
Correction = product = __.147 

Ans. True value of subchord = 45.147 


Example.—The last deflection at the end of a 40° curve is 
found to be 6° 80’. What are the nominal and true values of 


the last subchord? 
Here 4d = 6° 30’, and by eq. (82) A AT We 


0: _ 38.5 feet | 
20 
: ° an i 

Mrugrpabios «100 Tose 89 1008 feat Wi 


sin 20° wip 


Nominal value, ¢ = 100 


Or by Table VILI., excess of arc 40° = 2.060 
by above table, ratio for 32.5 feet = » .290 
Correction = product = _ .597 


Nominal value of subchord = 82.5 i i 
True value = 33.097 Wi 


108. For convenience in making deflections, the zeros of | 
the instrument should always be together when the line of at 
collimation coincides with a tangent to the curve. Thus, in 
beginning a curve, the transit being set at the P.C. zeros | 
together, and line of collimation on the tangent, the read- iI 
ing of the limb for any station on the curve has simply to be | 
made equal to the proper deflection from the tangent for that Hal | 
station. After the transit is moved forward from the P.C. NG so 
and set at another point of the curve, the vernier is set toa AVA 
reading equal to the reading used to establish that point, but | 
on the opposite side of the zero of the limb, and the line of A | 
collimation is set on the P.C. just left. Then by simply turn- Hi 
ing the zeros together again, the line of collimation will be an | 
made to coincide with a tangent to the curve through the new i 
point, and the deflections for the succeeding stations can be Pie 
read off directly, as before. Thus any number of transit | 
points may be used in locating a curve by finding the direc- Wh | 
tion of the tangent through each by a deflection from the pre- 1 
ceding point, until finally the P.7. is reached, where another 
deflection gives the direction of the located tangent. 


FIELD ENGINEERING. 


109. The assistant engineer keeps neat and systematic 
field-notes of all his operations with the transit in running 
curves. The numbers of the stations are written in regular 
order up the first column of the left-hand page of the field- 
book, using every line, or every other line, as may be pre- 
ferred. The second column contains the initials of each 
transit point on the same line as the number of its station, or 
between lines, if the point occurs between two stations, In 
the third column, and opposite the initials in the second, is 
recorded the station and plus distance, if any, of each transit 
point. The fourth column contains, opposite the ““P.C.,” the 
degree of curve used, and an R or Z, showing whether the 
curve deflects to the right or left; the fifth column contains 
the readings or deflections made from a tangent to set each 
Station or point, written on the same line as the number of 
that station or point; and the sixth column contains the cen- 
tral angle of the whole curve, A, written opposite the ‘‘ P. 7.” 
The plus distances recorded in 
the third column are always the 
nominal lengths of subchords, but 
if the true lengths have been calcu- 
lated and laid off on the ground, 
these should also be recorded in 
parenthesis. On the right-hand 
page are recorded the calculated 
bearings of the tangents and their 
magnetic bearings; and on the 
centre line of the page, opposite 

Fia. 7. the record of each transit point, a 

dot is made with a small circle 

around it, to show the relative position of the several points 
on the ground. Some slight topographical sketches may be 
made, indicating the more prominent objects, but the full 
sketches should be taken by the topographer in a separate book. 

110. Since the deflections start from zero at each new 
transit point, the sum of the deflections by which the transit 
points are located will be equal to one half the central 
angle of the curve. 

111. The stations on a curve may be located by deflee- 
tions only, without linear measurements. For this purpose 
two transits are set at two transit points on the curve, as A 


SIMPLE CURVES. 5Y 


and B, Fig. 7, and the proper deflections for any station are 
made with both instruments, the station being located by find: 
ing the intersection of the two lines of collimation. 

This method requires that the two transit points shall have 
been previously established, that their distance from each 
other shall be known, that they shall be visible from each 
other, and that they shail both command a view of the stations 
to be located. It is not therefore generally useful, but may 
be resorted to to set stations which fall where chaining cannot 
be accurately done, as in water or swamps. The chord join- 
ing the two transit points becomes, in fact, a base-line, and the 
deflections form a serics of triangulations. 

C. Location of Curves by Offsets. 

112. A curve may be located. by linear measurement only, 
without angular deflections. There are four general methods, 
Viz. : 

By offsets from the chords produced, 
By middle-ordinates, 

By offsets from the tangents, and 
By ordinates from a long chord. 


To locate a curve by offsets from the chords 
produced. 

When the curve begins and ends at a station. 

1138. Let A, Fig. 8, be the P. C. of a curve taken at a station, 
to locate the other stations, a, 0, ¢, 
etc. The chords Aa, ab, be, etc., 
each equal 100 feet, and since the 
angle AOa = D, the angle VAa = 
4D. (Tab. I. 20.) Taking an off- 
set ax =t, perpendicular to the 
tangent, we have in the. right- 
angled triangle Aza. 


az = Aa X sin }D 
or 


t =100 sin 4D (34) 
The offset ¢ is called the tangent 
offset, and its value is givenfor all 
degrees of curve in Tab. IV. col. 4. F1a. 8, 
If the curve were produced 
backward from A, 100 feet to station 2, the offset zy would 


60 FIELD ENGINEERING. 


equal ?t; and if the chord zA were produced 100 feet from A 
to a’, the offset a’ would also equal t. Therefore the distance 
aa’ = 2t,and the angle aAa' = D. So if we produce the chord 
Aa 100 feet to 0’, the distance bd’ = 2¢. 

To lay out the curve, stretch the chain from A, keeping the 
forward end at a perpendicular distance, t, from the line of the 
tangent to locate station a. Then find the point 0’ by stretch- 
ing the chain from @ in line with a and A, and then stretching 
the chain again from a, fix its forward end at a distance from 
b’ equal to 2¢. This gives station b. In the same way find 
other stations. 

When the last station, as d, of the curve is reached, produce 
the curve one station farther toe’. Then the tangent hee 
d is parallel to the chord ce", and laying off ¢ from ¢ and e" per- 
pendicular to this chord, the tangent c’e is found. If the work 
has been correctly done the tangent c"e will coincide with the 
given tangent VB. 

When the curve begins or ends with a subchord, 

114. Let ., Fig. 9, be the PC, and Ag the first sub- 
chord = ¢, and the angle VAa = 24, and let the offset aw = t,. 
Then 


t; = ¢ sin 4d (35) 

Producing the curve backward to the nearest station 2, we 
have another subchord Az = (100 — ce), and the angle y.Az = 4 
(D — d), and putting the offset yz = t, 

?, = (100 — c) sin } (D— d) (86) 

Laying off the two subchords on the ground, and making 
the proper offsets, ¢, and ¢,, at the 
same time, we fix the position of 
the two stations @ and z on the 
curve ; after which we may pro- 
duce the chord 2a 100 feet to b, 
and proceed as before until the 
curve is finished. 

If the curve ends with a sub- 
chord, as dB, produce the curve 
to the first station beyond B, as 
- e", then calculate the two offsets 
for the two subchords Bd and Be", 
and lay them off from d and e" 


SIMPLE CURVES. 


perpendicular to the supposed direction of the tangent. If 
the line de so obtained coincides with the given tangent, VB, 
the work is correct. 

115. We may find the values of ¢ and ¢, otherwise than 
by the formule above, for in Fig. 8 we have shown that the 
angle aAa' = aOA, and since these triangles are isosceles, 
they are similar; therefore 


Fig. 8, OA: Aa:: Aa: aa’ | 
or FR: 100::100 : 2¢ A | 
_ (100) | 
5 Fiore en 
and similarly, Fig. 9, | 
ce 

; = —— ( 

iso Bp (38) 
Hence 

Lee e= (1009 
roe. et 39 
= “q00P (39) 


Thus ¢, may be found by multiplying the square of the sub- ae 
chord by the value of ¢ given in Tab. LV., and dividing the At 
product by 10000. As c is always less than 100, so ¢, is always 
less than ¢. 

116. In eqs. (85), (88), and (89) it is customary to use the 
nominal values of c, and this can produce no error in ¢ or ¢, 
exceeding -005, when the degree of curve does not exceed ten 
degrees. In the case of a very sharp curve, the formule eqs. 
(40) and (41) are preferable. 


To locate a curve by middle-ordinates. 


When the curve begins and ends at a station. 

117. In Fig. 10, let A be the P.C. at a station, and let aand 
e be the next stations on the curve either way from A. Then, 
since zy = aw = ft, the chord za is parallel to the tangent A J, 
and Ag =—¢. Hence, having any two consecutive stations on 
the curve, as 2 and A, we may lay off the tangent offset ¢ 
from A to gon the radius, and find the next station, a, 100 feet Pe 
from A on the line zg produced. Then laying off ai = ¢ on Wt 
the radius @O, a point on the line Ah produced and 100 feet | 
from @ will be the next station 4. 


62 FIELD ENGINEERING. 


On reaching the end of the curve, the tangent is found 
precisely as described in the method by chords produced, § 113. 
In Fig. 10, we observe that if the radius 0A were unity, gA 
would be the versed sine of the angle a~0A = D.. But GA oat, 


hiss Rovers iD (4()) 


When the curve begins or ends with a subchord, 
118. Let A, Fig. 11, be the C,, and @ and z the nearest 


Fia. 10. Fie. 11, 


stations. Then Aa = ¢, the first subchord, and a@0A = d, and 
by analogy, we have from the last equation, if aw = t, and 
ey = t, 

¢, = R vers d 

t, = & vers (D—d) § (41) 


or eq. (389) may be used if preferred. 

Having found the two stations, a and z, on the curve, lay 
I off from the forward station a, ad = t on the radius, and go 
| continue the curve as described above. 

i When the end of the curve is reached, produce the curve to 
the next station beyond, and find the tangent by offsets as 
described in the previous method, § 114. 


To locate a curve by offsets from the tangents. 
When the curve begins at a station. - 


119. Let A, Fig. 12, be the PC. at a station. Then the 
next station @ is located by the tangent offset t, taken from 


SIMPLE CURVES. 63 


Tab. IV., or calculated by eq. (40). To calculate the distance 
anc offsets for the following stations, J, c, etc., in the diagram 
draw lines through the points 4, ¢, etc., parallel to the tangent 
AJ, intersecting the radius AOing’, g", etc., and draw the 
lines bz’, cz", etc., perpendicular to the tangent. 


Then 
Az’ = g'b = Ob sin BOA 


or 

Az’ = Ff sin 2D) 

Az" = Rsin 3D (42) 
and etc. etc. J 
Also, 

bz’ = g' A = Ob vers. DOA 

or 

t RR vera 2D 

i" = & vers 3D} (43) 
and etc. atCiins | 


But these calculations may be avoided, for as twice ag equals 
the chord of two stations, so twice dg’ equals the chord of four 
stations, and twice cg" the chord 
of six stations, etc. So also as Ag 
is the middle-ordinate of two sta- 
tion, Ag’ is the middle-ordinate of 
four, and Ag” the middle-ordinate 
of six stations, etc. Hence the 
rule: 

The distance on the tangent from 
the tangent point to the perpendicu- 
lar offset for the extremity of any 
are is equal to one half the long 
chord for twice that arc; and the 
offset from the tangent to the, ex- 
tremity of any are ts equal to the 
middle-ordinate of twice that are. 

The long chords and middle-ordinates may be taken from 
Tables VIL. and VIII. for 2, 4, 6, 8, etc., stations, when the 
P.G. is at a station, or for 1, 3, 5, 7, etc., stations, when the 
P.O. is at + 50, or half a station. 

If the offsets from the first tangent A V prove inconveniently 
long, the second half of the curve may be located from the 
other tangent BV, beginning at the point of tangent B, and 
closing on a station located from the first tangent. 


Fie, 12. 


64 FIELD ENGINEERING. 


When the curve begins with a subchord. 
120. If d=the angle at centre, subtended by the first 
subchord, we have for the distances on the tangent (Fig. 13) 


Az = RFR sind 
Az = Rsin d+ D) 


(44) 
Ax" = Ff sin (d + 2D) 
ete, etc. 
and for the offsets (Fig. 11) 
t, = vers d 
¢ = Rvers d+ D) t (45) 
t" = FR vers (d + 2D) 
etc. ete, 


If the first subchord equals 50 feet (nominal), then d =—1D, 
and the Tables VII. and VIII. may be used as explained 


Fie.13; Fie. 14, 


above. These tables may be used in any case, by adopting a 
temporary tangent through any station, and laying off the dis- 
tances on this, and making the offsets from it. 

When a curve is located by offsets the chain should be car- 
ried around the curve, if possible, to prove that the stations 
are 100 feet apart. , 

To locate a curve by ordinates from a long 

chord. 

When the curve begins and ends at a station. 

121. In Fig. 14 draw the long chord AB, joining the tan- 
gent points, and from this draw ordinates to all the stations on 


i 


SIMPLE CURVES. 65 


the curve. We then require to know the several distances on 
the long chord Aa’, at’, b'c’, etc., and the length of ordinate 
at each point. 

Let C =the long chord AB, then eq. (22 


C= 2R sinta 


If a is the second station and 7 next to the last on the curve, 
join az, and let the chord aa = C’. Then since the arc Aa = 
tk = D, the angle at the centre subtended by C’ is (A — 2D). 


*. MO=2KR sini (a — 2D) 
Again, if we join } and / the next stations and let b4= C" 
C" = 2K sin 4 (A — 4D) 
and so on for other chords. 
Since Aa’ = ki, C= C’ + 2Aa’ 
C—C 
2 


=o Ag = 


Similarly, : 
a b= aaa siked 
2 

Thus we continue to find the distances up to the middle of 
the curve, after which they repeat themselves in inverse 
order. 

122. When the long chord C, subtends an even number of 
stations (as 10 in Fig. 14), the middle ordinate of the chord is 
the ordinate of the middle station, ase. Since the chords AB 
and a# are parallel, the ordinate a'a or 7't is evidently equal to 
the difference of the middle ordinates of these chords. 

Let M, M’, M", etc., be the middle-ordinates of the chords 
CU, CU’, CO", etc. Then eq. (23) 


Me = vers A 
M' = Rvers§ (a —2D) 
M" = Rvers 4 (A —4D) 


etc., etc. 
And aa=ti=—=M-M 

b'd a h'h = MW — M" 

etc. ete. etc. 


The values of the chords and middle-ordinates may be taken 
at once from Tables VII. and VIII. 


66 FIELD ENGINEERING. 


Ezxample.—It is required to locate a 4 degree curve of ten 
stations by offsets from the long chord. 
By Table VIL: 


Diff. VDiff. 
10 sta. | GC =980.014 | 
: te Oh ap apa. | 1e0eaid | 95.205 =a 
eee) 194.059 | 97.080 2 ab = Th! 
6 « Oi = 595.744 is Ok 
£8 Crit 298. 7g0 | 196-962 | 98.481 = de = Hg 
ee rE a Aas 198.904 | 99.452 =e'd' =g'f 
ee a 199.878 99.939 = d'e' = fre’ 
0 | OG =000.000 | t 
From Table VIIL.: 
Diff. 
10st. | M =86.402 | | 
g « | Mi =55.500 | 30.902 =aa=v 
6 « M* =81.308 | 55.094 —)'b —Wh 
4 « | Mi = 18.948 | 72.459 —cc =9'g 
Q « | Miv— 3.490 | 82.912 —da= fF 
Q ‘ pea = 202000 S| 5/88; 40 hs aie 2 


123. When the long chord C subtends an odd number of 
stations, the middle ordinate will fall half-way between two 
stations, and need not be laid off. 

If the ordinates near the middle of the curve prove incon- 
veniently long, we may subtract 17 — M’, M—M", etc., and so 
obtain in Fig. 14 @’a, 6"b, ec, etc. We then lay off Aa’, aa, 
ab", b"b, be", etc., turning a right angle at every point. The 
chain should be carried along the curve at the same time to 
make the stations 100 feet apart. 

Example.—It is required to locate a 10-degree curve of nine 
stations by offsets from the long chord. 

By Table VII. : 


Diff. MDiff. 
9 sta. 811.814 
7 «© 658.105 153.209 76.604 = Aa’ 
5 484 900 173.205 86.603 = ab’ 
8 «' 996 969 187.938 93.969 ete. 
1 “ 400.000 196.962 98.481 
0 9.000... | 100.000 50.000 


4 


SIMPLE CURVES. 


By Table VIIL.: 


Diff. 
@ sta, 168.029 bie tts 
Saleen 50.000 =U) 
5 53. 80 | 34.202 = ¢"e 
Be rae TINORS oy 17.365 ete, 
1 2.183 | 4 Ha 
0 0.000 


124. The tables can be used equally well when the curve ill 
both begins and ends with a half station; also to locate 
half-station points throughout the curve, but in the latter case 
the numbers are taken from consecutive columns of the tables . 
instead of from alternate col- ah 
umns, as in the above examples. 


When the curve begins or ends 
with any subchord. 

125. Let A, Fig. 15, be the 
P.@%. and Aa=e the first sub- 
chord, and d the angle it sub- 
tends at the centre. In the dia- 
gram draw the long chord AB, 
and the ordinates to each sta- 
tion, and through each station 
draw a line parallel to AB, and 
let AOB= A. 

Since the angle VAB = 4A and Hi 
VAa = id, theangle aAB=4(A—d). The deflection angle li 
from the subchord Aa produced to the chord ad is (d+ D), te 
the deflection angle between any two consecutive chords of i 

i 
| 


100 feet is }(D+D)=D. Therefore the angle ae 
bab’ =4 (A — ad) — 4+ D) =F (a —2d-D) 
che” = 4 (A — 2d —D) — 3 (2D) =4 (a — 2d — 8D) 
edd’ = 4(A — 2d —8D) — + QD) =4(A — 24—5D) 

ete. 


etc. etc. 


FIELD ENGINEERING. 


Solving the several right-angled triangles we have, Fig. 15. 


Ad@=c. cos4(A —d@) ) 

ab" = 100 cos4(A —2d— D) | 

be" = 100 cos (A — 2d— 8D) } | (46) 
dd" = 100 cos i (A — 2d — 5D) 

etc., etc., | 

And also 

wa=c. sin¢(A —a@) 

b"b = 100 sint¢ (A —2d— D) 

¢ “ = 100 sin} (A — 2d — 3D) (47) 

= 100 sin 4 (A — 2d — 5D) 
BAL ete. 


When the middle point of the curve is passed the minus 
quantities in the parentheses become greater than A, making 
the parentheses negative, and, therefore, the sines negative, 
and indicating that such values as are determined by them 
ii must be laid off toward the long chord AB. 
| By a proper summation of the quantities determined by eqs. 
(46) and (47) we obtain the distances Aa’, Ab’, Ac’, etc., and 
the ordinates a‘a, b'b, c'c, etc., and the curve may be located 
accordingly. It is well to make all the necessary calculations 
before beginning to lay down the lines on the ground, thus 
avoiding confusion and mistakes. 

Example.—The P.C. of a 3° 20' curve is fixed at + 25 feet 
beyond a station, and the central angle is 16° 24’= a. It is 
' required to locate the curve by ordinates from the long chord. 
i We have c = 100 — 25 = 75 and d= 2° 30’ and D= 3° 20’. 
Ti Hence, eqs. (46) 


1 Aa'= %5 cos 6° 57 = 74.449 74.449 = Aa’ 
ir ab" = 100 cos 4° 02' = 99.752 174.201 = Ad’ 
be" = 100 cos 0° 42’ = 99.993 274.194 = Ae' 
a'd = 100 cos (— 2° 38’) = 99.894 374.088 = Ad’ 
e"e = 100 cos (— 5° 58’) = 99.458 473.546 = Ae’ 
éB= 17 cos (— 7 55’) = 16.8388 490.384 = AB 


By eqs. (47) 


va= Hsin 657 = 9.075 9.075 = wa 
6"6 = 100 sin 4° 02’ = 7.034 16.109 = b'b 
e"c = 100 sin 0° 42’ = 1.222 17.331 = ce 
ca" = 100 sin (— 2° 38) = — 4.594 12.737 = dd 
de" = 100 sin (— 5° 58’) = — 10.395 2.342 = ee 


ee’ = 17 sin (— Te 55’) ame 2.341 0.000 eo oe e 


q 


SIMPLE CURVES. 69 


The same formulz can be used when the curve begins at a 
station by making ec = 100 and d= D. 

126. The methods of locating curves by linear measure- 
ments do not require the use of a transit, although one may 
be used to advantage for giving true lines, turning right 
ungles, etc. When a transit is not used the alignments should 
be made across plumb-lines suspended over the exact points 
previously marked on top of the stakes. A right angle 
may easily be obtained, without an instrument, by laying off ATR 
on the ground the three sides of either of the right-angled 1 
triangles represented in the following table (or any multiples A a 
of them), always making the base coincide with the given line. i 


TABLE OF RigHt-ANGLED TRIANGLES. Ht 


Base. Hypothenuse. Perpendicular. i 
4 5 3 He 
12 13 5 | 
20 29 21 
24 25 7 
40 Al 9 
60 61 11 
84 85 13 


D. Obstacles to the Location of Curves. 

127. To locate a curve joining two tangents when the in- 
tersection Vis inaccessible. Fig. 16. 

From any transit point p on one tangent run a line pq to i 
intersect the other tangent; measure Hl 
pq and the angles it makes with the 
tangents. Then the sum of the de- 
flections at p and g equals the central 
angle A. Solve the triangle pyV 
and find Vp. Having decided on 
the radius R of the curve, calculate 
the tangent distance VA by eq. (21), 
and lay off from p the distance 
pA == VA — Vp to locate the point 
of curve. The point p being as- 
sumed at random, Vp may exceed VA, in which case the differ- itt 
ence pA is to be laid off toward V. 14 

In case obstacles prevent the direct alignment of any line | 
pg, a line of several courses may be substituted for it (as 


Fic. 16. 


7U0 FIELD ENGINEERING. 


explained in §§ 46, 47, 48,) from which the length of pq will 
be deduced. The algebraic sum of the several deflections will 
equal A. 


. 128. To locate a curve when the point of curve is 
inaccessible. Fig. 17, 
Assume any distance Ap on the curve which wil] reach to 
an accessible point p. Then by eq. (19) the angle 
Dx Ap 
100 
Ap' = R sin pOA 
pp = Rvers pOA 
Vp' = VA — Ap’ 


pOoA = 


Measure Vp’ and p’'p to locate a transit point at p; and meas. 

ure an equal offset from some transit point on the tangent, as 

q7. This gives a line pq’, parallel 

B to the tangent, from which deflect at 

: p an angle equal to pOA for the 

direction of a tangent through the 

point p. 

f, Instead of measuring the second 
Ip ' 

offset gq’ we may deflect from pq an 


U 


angle found by tan gpq’ = i and so 


BAS 


obtain the line pg’ parallel to the 


Fic. 17. tangent. Or we may deflect from pV 
the angle found by tan p Vp’ =F to obtain the line q'p pro- 


duced, from which the tangent to the curve at p is found as 
above. 

Again, we may lay off from V, the external distance VA 
found by eq. (24) or Tab. VI on a line bisecting the angle 
AVB. This gives us h, the middle point of the curve, and a 
line at right angles to 4V is tangent to the curve at h, from 
which the curve may be located in either direction. 


129. To locate a curve when both the Vertex and Point 
of curve are inaccessible. Fig. 18. 
From any point p on the tangent run a line pq’ to the other 


iq 


SIMPLE CURVES. 
tangent, and so determine pA as in §127. Suppose the curve 
produced backward to p’ on the perpendicular offset pp’. 

Then 

abe | CT, RIS 5 . 
sin p'OA = RP and pp’ = R vers p'OA 

Having located the point p’, a parallel chord p’g may be 
laid off, giving a point g on the curve, since p'g = 2 X pA. Hh | 
At g deflect from gp’ an angle equal to p’OA for a tangent to | 
the curve at q.. Wit 

If any obstacle prevents using the chord p’g, any other 


Fia. 19. 


chord as p's may be used, by deflecting from p’g the angle b 
gp's = 4 (gQs) and laying off its length, vi 


p's= 2R sin (p'OA + qp’s). ft 


At 3 a deflection from the chord sp’ of (p'OA + gp’'s) will give 
the tangent at s. 

If obstacles prevent the use of any chord, the methods de- 
scribed in $131 may be resorted to. 


130. To pass from a curve to the forward tangent when the 
Point of Tangent is inaccessible. Fig. 19. 

From any transit point p on the curve, near the end of the 
curve, run a chord parallel to the tangent. The middle point 
g of the chord will be on the radius through the point of tan- 
gent B. At any convenient point beyond this an offset equal 1 
to pp’ = R vers pOB may be made to the cangent, and at 
some other point an equal offset will fix the direction of the 
tangent. 


22 FIELD ENGINEERING. 


Otherwise, if an unobstructed line pq can be found inter. 
secting the tangent at a reasonable distance from B, measure 
the angle q'pg = pap’, and lay off the distance 


r 


/ Pp 


sin q'pq 
to fix the point g. Then 
Ba=p'¢ —p'B= pp’ cot q'pg — R sin pOB. 


Otherwise ; assume an arc of any number of stations from 
p to q" on the curve produced, and take the length of chord 
from Tab. VII. Lay off pg’, and from q’ lay off g'g=R 
vers g"OB, perpendicular to the tangent, to locate g. The 
angle pg"¢ = 90° — q'pq", and the distance qB = R sin gq" OB. 


131. To pass an obstacle on a curve. Fig. 20. 

From any transit point A’ on the curve take the direction 
of a long chord which will miss the obstacle, as A'B’. The 
length of this chord is 2R sin 

V'A'B', V'A' being tangent to the 

curve at A’ (see eq. 22), and by 
measuring this distance, the point 
B' on the curve is obtained. If 
the angle V'A'B' is made equal to 
the deflection for an exact number 
of stations, the chord may be taken 
from Tab. VII. 

If the chord which will clear the 
obstacles would be too long for con- 
venience, as A'g’, we may measure 
a part of it as A’p’, and then, by an 
ordinate to some station, regain the curve at p. The distance 
on the curve from A' to p being assumed, the distances A'p’ 
and p’'p are calculated by the methods given in § 121 to § 125. 
If p'’p can be made a middle ordinate the work will be much 
simplified. If more convenient the middle ordinate may first 
be laid off from A’ to p", and the half chord afterwards 
measured from p" to locate p. 

Again, we may calculate the auxiliary tangent A’V’ for 
any assumed length of curve A'B’, and lay off the distance 
A'V’ and V'B’, deflecting at V’ an angle equal to twice 


Fia. 20. 


SIMPLE CURVES. 


V'A'B'. But if the point V’ should prove inaccessible, we 
may conceive the auxiliary tangents to be revolved about the 
chord A'B' as an axis, so that V’ will fall at V", and the 
lines .4'V" and V"B' may be laid out accordingly. If these 
in turn meet obstructions, we may run a curve from A’ to B' 
of same radius as the given curve, but tangent to A’V" and 
VB’. 

Again, the entire curve or any portion of it may be laid out 
by offsets from the tangents, or by ordinates from a long 
chord, as already explained, § 119 to $126. 

In case any distance on a curve must be measured by a tri- 
angulation, as in crossing a stream, a long chord may be 
chosen, either end of which is accessible, and the triangula- 
tion is then performed with respect to this chord or a part of 
it, as upon any other straight line. 


SPECIAL PROBLEMS IN SIMPLE CURVES. 


132. Given: a curve joining twotangents, to find the change 
required in the radius BR, and external distance EK, for an 
assumed change in the value of the tangent distance'T. Fig. 21. 


Fia. 21. 


Let.7 =AV= VB and 7’ = A'V= VB' 
ih = AO “« R'=A'0,' 
ery == VE. +h SV A 


Then 7’— 7’ = AA' = the given change. 


By eq. (20) Kk =T. cot4a 
R'= T' cotta 


0G =R—R'=(T—T')cot4a 


74. FIELD ENGINEERING. 
By eq. (26), similarly, 
HH' = H— E'=(T—T’') tania (49) 


Eqs. (48) (49) give the changes in R and EF for any change 
in 7. When 7'is increased R and # will be increased also, 
and vice versa. 

Example.—A 4° curve joins two tangents, making an angle 
of 388° = A, and it is necessary to shorten the last tangent dis- 
tance 80 feet. What will be the change in tne radius and in 
the external distance? 


Eq. (48) T—T'=80 log 1.908090 
$A 19° log cot 0.463028 
i | Ans. R —R' 232.34 log. 2.366118 
} R 1482.69 
Riss 1200.35 or about 4° 46’ = D’. 


If the tangent distance had been increased 80 feet we should 
add the above to R. 


ft’ = 1665.03 or about 3° 26’ = D' 


| Eq. (49) 7-7’ =80 log 1.903090 
| +A 9° 30’ log tan 9.223607 
| Ans. E—E' 18.387 log 1.126697 


133. Given: a curve joining two tangents, to Jind the change 

required tn the radius R, and tangent distance T, for any 
| | | assumed change tn the value of the eaternal distance B. Fig. 21. 
i We suppose HH’ given to find OG and AA’. 


By eq. (24) H =R exsecta 
E' = R' ex sec4+a 
OG SR eRe 


ex sec dA 


By eq. (49) 
AA = T —T' =(H— FE’) cota 


SIMPLE CURVES. 75 


Example.—A 4° curve joins two tangents, making an angle 
of 38° = A, and it is necessary to bring the middle point of 
the curve 25 feet nearer the vertex V. What changes are re- 
quired in the radius and point of curve? 


Kq. (50) H-—H'= 25 log 1.397940 

tA 19° log ex sec. 8.760578 

Ans. R—R' 483.87 log 2.637362 
R 1482.69 


R’' 998.82 or about 5° 44’ = D’ 


Eq. (51) H- £#' 25 log 1.397940 
tA 9° 30 log cot 0.776393 
T — T' 149.39 2.174333 


or the P.C. will be moved toward the vertex 149.39 feet. 
But if the point H, Fig. 21, were to be moved 26 feet 
further. from the vertex V, then 


R' = 1866.56 or about 38° 04’ = D' 


and the P.C. will be moved 149.39 feet further from the 
vertex. 

It is preferable to assume some radius from Table IV. near 
the value of R’ found as above, and from this calculate the 
value of 7" by eq. (21). 


134. Given: a curve joining two tangents, to find the change 
made in the tangent distance 'T, and external distance KH, by 
any assumed change in the value of the radius R. Fig. 21. 


By eq. (48) 

AA'=T—T'=(R—#') tanta (52) 
By eq. (50) 

HH'=H—H'=(R— R')ex sec4a (53) 


The changes calculated by eqs. (52) (53) will be added to or 
subtracted from 7 and # respectively, according as the radius 
ie increased or diminished. 

135. Since for a constant value of the central angle A, 


%6 FIELD ENGINEERING, 


the homologous parts of any two curves are proportional to 
each other, we may write at once 


oon | tee nat ated ane 

SECA’ iabahtt ik yeh 
ee 2 

a IT ps : ny Uae M' 

Let Reet Lea ace 

etc. etc. ete. 


1356. Given: a curve joining two tangents, to change the 
position of the Point of curve so that the curve may end 
in a parallel tangent. Fig. 22. 

Let AB be the given curve, AV, VB the tangents, and 
V'B' the parallel tangent. Then VV" is the distance from 
one vertex to the other; and since 
there is no change in the form or 
dimensions of the curve, we may 
conceive it to be moved bodily, 
parallel to the line AV, until it 
touches the line V'’B’, when every 
point of the curve will have moved 
a distance equal to VV'. Hence 
AA'=00'= BB'=VV". ‘There- 
fore, run a line from B parallel to 

Fia. 22. pe a4 Vi intersecting the new tangent in 

B’, measure BB’, and lay off the dis- 

tance from A to find A’. In the figure the new tangent is 

taken outside the curve, and so A’ falls beyond A, but if the 

new tangent were taken inside the curve at V"B", the new 
P.C. would fall back of A at some point A’, 

If the parallel tangent is defined by a perpendicular offset 
from B, as Bp; since the angle BB'p= A 


AA' = BB’ =~ (55) 


137. Given: a curve joining two tangents, to find the 
radius of a curve that, Srom the same Point of curve, will end 
na parallel tangent. Fig. 28. 

Let AB be the given curve, AV, VB the tangents, and 
V'B' the parallel tangent; and let AO= Rand AO’ = R’. 


mw 


SIMPLE OCURVES. var 


Since the central angle A remains unchanged, the angle 
4A between the tangent and long chord remains unchanged; 
therefore V'A B' = VAB, and the new point of tangent is on 
the long chord AB produced. Find on the ground the inter- 
section of V'B’ with AB produced 
and measure BB’. In the diagram 
draw Be parallel to AO, then Beb' = 
A, and by eq. (22) 


BB’ = 2Be sin 4A 


but 
Be = 00' — ee a te 
BB ~ 
at ey 2sin tA 8) 0 
The + sign is used when B’ is be- Fig, 23. 


yond B, as in the figure; but if the 
parallel tangent is within the given curve it will cut the 
chord in some point B", and then the — sign must be used, 
since R' will evidently be less than Lt. 

If the parallel tangent is defined by a perpendicular offset, 
as Bp = B'f; since Beb' = A 


Bp = Be vers A =(h' — f) vers A 
a Ri= k-- ——— (57) 


Add or subtract as explained above. 
If the long chord C= AB is known, then the new long 
chord C' = AB' or AB" =C + BB’, and by eq. (54) 


pi = pO * BB ne 


138. Given: a curve joining two tangents, to change the 
radius, and also the Point of curve, so that the new curve 
may end in a parallel tangent directly opposite 
the given Point of tangent. Fig. 24. 

Let AB be the given curve, AV, VB the tangents, V'B’' the 
parallel tangent, and B' the given tangent point on the radius 
OB produced. 


18 FIELD ENGINEERING, 


In the diagram, produce the tangent AV and the radius OB 
to intersect at K. Then 


BK = R exsec A 
B'K = R' exsec A 


Subtracting we have 


BB' =(R — R') exsec A 


ae (59) 


Fig. 24, Fig. 25. 


To find the change AA’ of the P’ C., in the diagram draw 
O'G parallel to _A’A; then 
O'G = OG tan A . 
or 
AA'=(R— RP’) tan a (60) 


Vil By substituting the value of (R — R') from eq. (59) and ob- 
serving Table II. 42 we have 


AA' = BB' X cot ta (61) 
Observe that. eqs. (59), (60), and (61) may be derived directly 


from eqs. (50), (52), and (51) respectively by writing A for $A. 


139. Given: a curve Joining two tangents, to find the new 
tangent points after each tangent has been moved 
parallel to itself any distance in either direction. Fig. 25. 


SIMPLE CURVES. 79 


Let A and B be the given tangent points, and A’ and B’ 
the new tangent points required. Let the known perpendicu- 
lar distances Ag=a, and Byp=b. We then require the 
unknown parallel distances gA’ = @ and pB' = y. 

Since the form and dimensions of the curve remain un- 
changed we may conceive the curve to be moved bodily 
into its new position on lines parallel and equal to the. 
line VV' joining the vertices. Then AA’ 00'= BB'= 
Be 

In the diagram draw VK parallel and equal to Bp = 6 and 

V'H parallel and equal to A4g=a. Then VH=gA'=2, and 
V'K=Bp=y. Since VGV' = JA, we have 


VG=- and GH = e 

sin A tan A 

and since 
VH = VG—GH=z 

* b pe AER 

~ gin A tan A 
Similarly (62) 

b a 


Y=Tan A sin A 


When the new tangents are outside of the given curve, the 
offsets a and } are considered positive; if either new tangent 
were inside of the given curve its 
offset would be considered negative. 
In solving eqs. (62) if 2 and y are 
found to be positive they are to be 
laid off forwards from q and 7, as 
in Fig. 25; if either is found to be 
negative it is to be laid off in the 
opposite direction. 

Example.—A certain curve has a 
central angle of 50° = A, and it is 
proposed to move the first tangent Fic, 26. 
in 20 feet and the second tangent 
out 12 feet. Required, the distances on the tangents from the 
old tangent points to the new, Fig. 26. 


rs 


80 FIELD ENGINEERING. 


Here a = — 20 andd’=-+ 12 


4b 12 1.079181;—a 20 1.301030 
A 50° —_ log sin 9.884254| A 50° ~_—— dog tan 0.076186 


15.665 1.194927; -— 16.782 1.224844 
x = 15.665 — (— 16.782) = + 82.450 


+2 12 1.079181 |—a 20 1.301030 
wie b0° log tan 0.076186 | A 50° log sin 9.884254 


10.069 1.002995 | — 26.108 1.416776 


y = 10.069 — (— 26,108) = + 36.177 


3 (&@ = — 82.450 

For + a and — 0d ly = — 36.177 
g=— 1,120 

For + a and + 0 ang 
fh by fee + 1.120 

For — a and — 6 j y = + 15.989 


If we have a and @ given to find d and y: Solving eqs. (62) 
for 6 and y we obtain 


b=asin A—+-acos ait 


(63) 
Y¥Y=2cOS A —a@sin A 


In which the algebraic signs of the quantities must be ob- 
served as above. 


140. Given: a curve joining two tangents, to find a new 
Radius and new position of the Point of curve, such 


i that the curve may end at the same point as before, but with 


a given change in the direction of .the forward tangent. 


| Fig. 27. 


Let AB be the given curve, AV, VB the given tangents, 
V’B the new tangent, and VBV' the given change in direc- 
tion. Let A’= A+ VBYV'". 


a | 


SIMPLE CURVES, 81 


In the diagram draw BG perpendicular to AV produced; 


then 
BG = R vers a 
<= F' vers A 
Hence 
: vers A 
R= Re (64) 
vers A 
and 


AA =AG— A'G=Rsin A—R'sin a’ (65) Hie 


In the figure the change in direction of tangent makes A' 
greater than A; therefore V' falls beyond V, and A’ beyond 


Fie, 27. Fira. 28. 


A; but if the change made A’ less than A, then VY" and A’ | 
would fall behind V and A respectively, and R' would be We 
greater than R. 

The same formule apply to the converse problem in which | 
B is taken as the point of curve, and A and A’ as points of HN | ig 
tangent. HW 

| | 


141. Given a curve joining two tangents, to find the change 
tn the Point of curve when the forward tangent takes a new WW 
direction from ihe vertex V. Fig. 28. fl 

By eq. (21) 1 

¥ ; eH 


VA=RianjaA, VA' = Rtan $A’ 


AA' = R (tan 4A — tan $A’ (68) ii 


142. Given: a curve joining two tangenis, to find the new 


f: 


82 FIELD ENGINEERING. 


radius, R, when the forward tangent takes a new direc- 
tion from the vertex, V. Fig. 29. 
By eqs. (21) (25) 


VA=Rtania, R'=VA cot ta’ 
K' = R tan 4a cot 4a’ (67) 


143. Given: a curve joining two tangents, and a given 
change in the direction of the forward tangent from the 
verted, to find the radius and point of curve of a curte 
that shall pass at the same distance, VHA, from th. vertex. 
Fig. 30. : 

Let AB be the given curve, BVB’ the given change in 


i Fie. 29. Fie. 30. 


direction of tangent, and VW’ = VH. Let a’ = a+ BVB', 
then eq. (24) 


1) VH = Rex sec £4 = VH' = R' ex sec $4’ 


| es R= R. LS8ee eS (68) 
ii exsec $A 
i By eq. (28) 
VA=VHcot4a, VA' = VH' cotta’ 
ge AA' = VH (cot +4 — cot $4’) (69) 


But in case A’= A — BVB', AA’ becomes negative and 
must be laid off backward from A, 


SIMPLE CURVES. 83 


Example.—Given a 2° curve, A = 80° and BVB' = — 10° 


aie a a ( 
R log 3.457114 
tA 40° . log exsec 9.484879 
VH 874.97 2.941993 
tA! 35° log exsec 9.348949 
Re 1° 27’ nearly 3.598044 
tA 20° cot 2.74748 
tA’ 17° 30 cot 3.17159 


— 0.42411 


AA' = 874.97 X (— .42411) = — 371.08 


and must be laid off backward from A. 


144. Given: two indefinite tangents, a point situated be- 
tween them, and the angle A, to find the radius R, and tan- 
gent distance T of a curve joining the tangents which shall pass 
through the given point. Fig. 31. 

If the given point is on the bisecting line VO, as H, meas- 
ure VH = #, and find #& and 7’as in §§ 97, 98. 

When the given point, as P is not on the bisecting line VO; 
if a line GK is passed through P per- 
pendicular to VO, it will be parallel 
to any long chord, as AB, and the 
angle V@K=+4A. The curve pass- 
ing through P will intersect GX in 
some other point P’; the line GK 
is bisected by the line VO at J, and 
Wid oeey 2A fh 

If the given point P is located by a 
perpendicular offset from the tangent, 

RS "as PL; in the triangle PLG, LG = 
PL cotta. Lay off LG, and at G deflect VGK = 4A, and 


measure GP and PK. Since by Geom. (Tab. I. 24) GA? = 
GP' x GP, and GP’ = PK; 


GA= VGPX PK (70) 


84 FIELD ENGINEERING. 


Lay off GA; and A is the Point of curve, AV= 7" and 
k= AV cotta. 

If the given point were located by an offset from BV. find 
B first, and make VA = BV. ' 

If the given point P is located by a perpendicular offset 
JP from the bisecting line VO; produce JP to intersect the 
tangent at G and measure PG. Since P'G = GP + 2PI 


GA= VGP(G@P+ 2PT) (71) 
whence we have the point of curve A, as before. 


145. Given: a curve, AP, and the radial offset PP’ 
Al to find a curve which shall pass through the point P’, start- 
Hi ing from the same point of curve A. Fig. 32. 


Fia. 82. 


| Let 6 = PP’, and in the diagram draw P’'@' parallel to the 
| common tangent 4X, and join AP’. Then 


P'@ =(R + d)sin a 

GA =R—-(R + b)cosa 

1} | rE yom ry: 

i | ig tan $A pie (Rot) A (72) 
if P' / Ss] 

| Rim 2%, (2 + Deine (73) 
sin A sin A 


When the offset is outward use R -+ 6, when it is inward 
use R — 0, 

Haample.—Given: a 8° curve of 16 stations and a radial 
offset of 205 feet inward from the P. 7. to find the radius of 
the curve passing through the extremity of the offset, 


4 


CURVES. 


SIMPLE 


Here A = 3° X 16 = 48°; and b = 205. 
R 3°= 1910.08 


R—b 1705.08 log 3.231745 
A 48° log sin 9.871073 
Sig day 3.102818 
RR? log 3.281051 

1.50742 0.178283 
A 48° cot .90040 
ZA’ tan .60702 = 31° 153’ 

2 

LS 62° 3L’ log sin 9.947995 
LP. log 3.102818 
R' (about 4° 01’). Ans. 3.154823 


If the same offset were made outside of the curve we should 
find R' log 3.488350, or about a 2° 05° curve. 

This solution is inconveniently long for ordinary field prac- 
tice. When the offset is small compared with the length of 
curve, we may use the following 

Approximate Rule: Divide twice the offset b by the 
length of curve, look for the quotient in the table of nat. 
sines, and take out the corresponding angle, which multiply by 
100, and divide by the length of curve. The quotient is the 
correction for the given degree of curve; to be subtracted when 
the offset is made owtward, and added when the offset is made 
inward. 

This rule is expressed by the formula 


100 gna od 
D'=D*F —F- sin 3 (74) 
Taking the same example, we have 
20 = sin 14° 51’ 
Ty = sin 


: ee acy OU * eal 
and correction = 14° 51’ < 1600 — = F 0° 56 


Hence D' = 3° 56’ or D’ = 2° 04 


FIELD ENGINEERING. 


THE VALVOID. 


146. Given: any number of circular curves of equal length 
Li, all starting from a common point of curve A, in a@ common 
tangent AX, to find the equation of the curve joining 
ther extremities, Fig. 33. 

Let AP be any one of the given curves, 

‘“ R= its radius AO, 

“ D = its degree of curve, 

“ A = its central angle AOP, 

*“‘ C= its long chord AP. 


Fig. 33. 


By substituting the value of 2 from eq. (16) in eq. (22) we 
have 
sin tA 


a ye +D 


(75) 


Substituting in this the value of D from eq. (20) and letting 


C L 
th t 6 = = —— 9 oe 
(theta) 4A, (rho) p 700 and VV {00° we have for the 
polar equation of the required curve 
sin 9 
sin NV 


in which p is the radius-vector AP, 6 the variable angle 
XAP, the unit of measure is one side of the inscribed polygon 
by which the circular curve AP is measured, and JV the num. 
ber of these sides in the length of the curve AP. By the 


1 


SIMPLE CURVES. 87 


conditions of the problem JV is constant, but 6 may have any 
value whatever. If we let 9 vary from 0° to + 180° and from 
0°. to — 180° the point X will describe the curve XP PA 
shown in the figure, which is called the Valvoid from its re- 
semblance to the shell of a bivalve. All circular curves tan- 
gent to AX at A and having a length IL = AX will terminate 
in the valvoid, and the line PP’ joining the extremities of 
any two of them is a chord of the valvoid. 


147. To find a tangent to the valvoid at any point 
P. Fig. 34. See Appendix. 
Differentiating eq. (76) 
dp ( if 6 
ST Ret & cot 6 — +, cot =) (77) 
which is essentially negative, since p is a decreasing function 
of 9. 
Let (phi) p = APG, the angle between the radius vector 
and the normal PG. 
1 6 
tan @ = a cot WoT cot 6 (78) 
The line PK perpendicular to P@ is tangent to the valvoid 
at P, and PV perpendicular to PO is tangent to the curve ALF, 
Then APV =9 and VPG =6 — 9g, and letting 1 = OPK = 
VPG. 
i=0-p=}A-— (79) 
Therefore, to obtain the direction of a tangent to the val- 
void at any point P, deflect from 
the radius PO an angle equal to 
i=(tA — @), on the side of PO 
farthest from the point ot curve A. 
The value of 7 may be found by 
eqs. (78) (79), but we are saved 
this somewhat tedious calculation 
by the use of Table X. 1, which K 


Fie. 34. 


wht 
contains values of the ratio ve U 
for various values of A, and length of curve L. Multiplying 
A by the proper tabulated number gives the value of 7 = OPK 


at once; or 
i=(4A —Q)=UA (80) 


88 FIELD ENGINEERING. 


148. To find the radius of curvature of -the valvoid 
atany point P. See Appendix, 
Differentiating eq. (77) we have 


dp | 2 6 1 ( 6 
ae poe tgs ie §80b woot 71.9 Gott 
ie =P 1 yr Cot G08 sachs cotta; + 1) 


The general formula for the radius 


curves is 
8 
2 mer 
(c T ape 
ee 


a p? pie 9 dp? ap 


do? ? age 


of curvature of polar 


, 


SUT, witty : d 2 
Substituting in this the values of p, 7 and fia and putting 


6 
(7 cot a te cot a) = @ we have after reduction, 


p (1+ a3 
Y bel —— DY e ne eae ee SS ELE Se 


: (81) 
1 — a2 4% cot 6 
This formula bein 


g too complicated for convenient use in 
‘he field, its use is 


avoided by referring to Table X. 2, which 


: Pitas : 
contains values of the ratio Er % for various values of A and 


L. Multiplying the given value of Z 
ratio, gives the value of the radius of c 
for a short distance either way from th 


by the proper tabular 
urvature of the valvoid 
e given point P; or, 


PEEVE, Tu (82) 


149. To find the length of are of the valooid corre- 
sponding to a change of one degree in the value of the 
angle A. Fig, 35, 

From any chord AP suppose a deflection of 4 degree to be 
made each way to Ap' and Ap"; then the angle p’ Ap" = 4° — 
the change in 6, and since A — 26, this makes a change of 1° 
in the value of A. We then require to know the length of 


SIMPLE CURVES. 89 
the arc p'p", and we may, without sensible error, consider it 


to be described by the radius of curvature r= Po for the 
point P, through an angle p’op". Now 


and since g’ is so nearly equal to m" we may assume uw’ = 
" ! tw Nie A" ‘ rte 
Meet RENCE .Q — Dip == re (1 —2w) and p'op" = 
4 a 
(A'— Aty (tu). 
But the condition of the problem requires A'’— A” =1’, 
hence p'op" = (1 —u)°. 
Therefore the length of arc p'p" for a change of 1° in the 
value of A is 


1=r(i—wu) X arc! 
or (Tab. XVII.) L=r (1 — wu) .0174538 
and since 7 = voL (Tab. X. 2), 
1 =v(1 — wu) LZ .0174533 (83) 


By this formula Table X. 3 has been prepared, for various 
values of A and L. 


150. Given: two curves of the same length L but of 
different radit, starting from the same point of curve in @ 


90 FIELD ENGINEERING. 


common tangent, to determine the direction and length of 
alne joining their extremities. Fig. 36. 

Let AX be the common tangent, and AP’, AP" the two 
curves, to determine the direction and length of D a iid 


If we take the point P on the 
are P'P" determined by the 
cr! airs ae 


angle A = > and draw 


a tangent PK to the valvoid at 
P, we may assume without ma- 
terial error that the chord P'P" 
will be parallel to PK for any 
value of P’P" not exceeding 
4Z, a limit not likely to be ex 
| ; ceeded in practice. 

| Let O be the centre of the curve AP fixing the point P; 


then AOP — at Pel rae 


OP =t= ae. 


Since P’P" is assumed parallel to PE 
: PPO = K@0' mata Ka n'— ATA Gg ay) 


P'P'O' — 7 — A Se A (1.46) (84) 


Similarly producing P"P’ to any point H, 


HPO =~ Ma ae rn har (85) 


a 


| whence also 
t= 2 = At A" (85)' 
The slight error involved in the above assumption is cor. 
rected by taking out the value of w (Table X. 1) correspond- 


ing to A", the less of the two given central angles; we have 


therefore written 7% with the double accent in equations (84) 
and (85). 


i 
SIMPLE CURVES. 91 


When 7’ and 2” are positive, they will be deflected as in 
Fig. 36, on the side of the radius farthest from A ; should 2" be 
negative it will of course be deflected from P’O" toward A. 

The arc P’P" corresponds to a change of the central angle 
from A’ to A" ; hence 


A Ar = A" ite CA BERS 
br 
PT ea ie S20 Nn ely (86) 


in which 7, is taken from Table X. 3 for Z= AP, and 
ee 
eS 


A em 


As in practice, the distance P’P" is usually small compared 
with Z, the arc and chord will be almost identical and no 
further calculation is necessary. If P'P" is large, it will be 
found that equation (86) gives the “ength of arc very correctly 
when a does not exceed, 20°, and the length of chord 


A'+ aA" 
2 


gives a value to P’P", between that of the arc and chord. 
The arc P'P" may be considered bu be described by the radius 


60° ; tor intermediate mean angles it 


when 


y = vL, v being taken fox tit Eas Ay (Table X. 2), and its total 


curvature is foune »y 4 ea re its length by the degree of 
curve corres,onding to r (Table IV). 

Example. Given, a 2° 30' curve, and a 1° curve of 12 stations 
each from the same PC, to determine the distance between 


their extremities. 


A’ = 24° x 12 = 80°, Aaueeaiees a = 21; 
oe ae BS 1 u" = .83446 

Kq. (84). 2” = 2°.97387 = 2°58'25" 

Eig. (85). ¢7 = 2" + A'—- A= 20°.9737 = 20°58'25" 

Eq. (86). Arc P'P" = 18° X 10.425 = 187.65 ft. Ans. 


Eq. (82). 7 = 1200 x .7479 = 897.48 ft. = (say) a 6°23’ curve. 

Total curvature, P’P” = 6°.883 x 1.8765 = 11°.9777. 

(The distance P’P"” may be found by solving the triangle 
formed by itself and the long chords of the curves APS 


AP".) 


FIELD ENGINEERING, 


151. Given: a curve AP, to Jind a curve starting from the 
same point A, that shall shift the station P any desired dis- 
tance PP’ to the right or left. Fig. 36. 

Before we can determine what distance PP’ is desired, we 
must know (approximately) its direction. We have given, 
therefore, D, LZ, and A to find the angle OPP’, and (after 
measuring PP’) to find A’ and D’. 

The solution is necessarily somewhat approximate, yet 
Close enough for all practical purposes. For if the required 
value of D’ were obtained precisely, it would probably involve 
some seconds, and would therefore be discarded in favor of 
some value in even minutes. 

When P’ is inside the given curve : 


Vi Eq. (80). t= OPK =ua. TableX.1 
H 
Eq. (82). r=Po =v. Table X. 2. 


Let 6 (delta) = degree of curve corresponding to 7, by 


Table IV. 
| ; atese Sieg A ad tied” : 
| <3 OP sy [00 40 nearly. 
Eq. (86). A’= A+ ae Table X. 38. 


4 


Instead of taking 2 from Table X. 3 for the exact value 


of A it is well to take it for the estimated value of AA, 


Eq. (20). D= Al 


When P’ is outside of the given curve: 


t= WA, 7 == OTR 


180° — OPP’ = f+ ei - 45 nearly. 


A *100 88 


Heample. Given, a 4° curve of 800 feet, or A = 82° to find 


SIMPLE CURVES. 93 


« curve from the same P.@. which shall shift the last station, 
in, about 55 feet. (Fig. 36.) 
i = 32° & .8355 = 10°.786 
7 = 800 X .7450 = 596, -. 6 = 9° 36’ = 9°.6 


geod ono D0 ° +S 9° U 
OPP' = 10°.736 iain 06 

[) ae ° 19) = % ° 

A' = 82°-+ 53, = 40 


1 = 40 =— 5°. Ans. 


For a 5° curve, the true distance PP’ = 59.53 
ie RRS 4°59' ‘ce ée¢ ‘ec 6 PP' = 54.60 


which proves this method practically correct. 


152. Given: a tangent and curve, and a straight line 
inter seotins them, making a given angle with the tangent at 
a given point, to determine the distance on the line 


from the tangent to the curve. Fig. 3%. 


oO T 
Fia. 37. 


We have OA, AG, and the angle AGP to find GP. 
R 


tan AGO = iG PGO = AGO — AGP 
Og sin PG-O 
sin OPT = 73 sin PGO = in AGO 


_ 4 sin (OPT— P@O) 
PG = BR sin PGO- 


94 FIELD ENGINEERING, 
When AGP = AGO, eq. (24), 
GP = R exsec (90° — AGO) 
When AGP = 90°, $8 (92), (119), 
GA 


sin POA‘ —— 


GP= R vers POA, R 


When AGP'> AGO, we have 
P'GO = AGP' — AGO 
but the other formule remain unchanged. 


Hxample.—Let R= 955.37, AG = 350, AGP = 40° 


R 955.37 log 2.980170 
AG 350. log 2.544068 
AGO 69° 52’ 47” log tan 0.436102 
AGP. 40° 

PGO 29° 52! 47" 


log sin 9.697387 


AGO 69° 52’ 47” log sin 9.972653 


OPI 32° 02’ 86” log sin 9.724734 
POG 2° 09' 49” log sin 8.576953 

8.879566 
R log 2.980170 
PG 72.40 Ans. 1.859736 


log 1.859736 


This problem may be used in passing from a tangent to a 
curve when the tangent point is obstructed. The distance 
AP on the curve is defined by the angle AOP, which is readily 
found. 


If AGP' > 2AGO the line will not cut the curve. 


153. Given: a curve and a distant point to find a 
tangent that shall pass through the point. Fig. 38. 

We have the curve adg and the point P visible, but distance 
unknown, to find the point of tangent B. 


SIMPLE CURVES. 


Any chord, as Of, parallel to the required tangent, if pro- 

duced will pass the point Pata perpendicular distance equal 
to the middle ordinate of that chord. Ranging across every 
two consecutive stakes on the curve we at first find the 
range falling outside of the required tangent, as bcG, ca H, 
etc.; but finally the range falls inside, as dek. We then know 
that the required point is between ¢ and é. 
If the range ce falls inside the point Pia , 
perpendicular distance equal to the middle Ki 
ordinate of ce, the tangent point is at d. 
If the perpendicular distance is greater 
than this, the point B is between c and d. 
If less, or if the range ce falls outside of 
P, the point B is between d and ¢. The 
middle ordinate for ce (200 feet) equals the 
tangent offset for 100 feet, given in Tab. 
IV., and it is generally so small that it can 
be estimated at P without going to lay 
it off. 

To find the exact point B, when it falls 
between d and e¢, find by trial a point x 
on the arc cd in range with ¢e and a point 
inside of P a perpendicular distance equal 
to the middle ordinate of ew. The point B 
is at the middle point of the arc ew If 
the point B is between ¢ and d, stand at ¢ 
and find a point 2 on the arc de in the same 
way. Bis at one half the arc cz. ; 

The middle ordinate of any chord ez is Fig. 38. 
less than VM for 200 feet, and greater than m for 100 feet. If 
necessary, its exact value m’ can be found by 


pos eee = = 


Wye PN Non 


= "70000 tt 


7 


‘and*this equation is nearly true when ea is as great at 300 or 
400 feet. That is, middle ordinates on the same curve are to 
each other as the squares of their chords very nearly. 

By this method the point B is found without the use of the 
transit, so that the plug can be driven at D before the transit 


to be near the unknown tangent point B 


96 FIELD ENGINEERING. 


is brought up from the rear. It is therefore preferable to the 
following solution. Fig. 39. 

From any two points @ and c¢ of the curve measure the 
angles to the point P, so that with the chord ac asa base, 
and the measured angles, we may find cP by the formula 


sin caP 
cP = ac — 


sin 2Pa 


Knowing the angle c that cP makes with a tangent at c, we 
find the length of the chord cd by cd = 2R sinc. 
By Geom. Tab. I. 24, 


PB = Pe= V¢eP X dP 


whence we know ce. Opposite e, or on the arc eB described 
With the radius Pe, we find B. 


Fia. 40, 


154. Given: two curves exterior to cach other, to 
Jind the tangent points of a line tangent to both and its 


length between tangent points. Fig. 40. 


Let B and A be the required tangent points. Let OB = BR, 


and 0'A = R’, 


On the curve of greater radius R select a point H supposed 
, and knowing the 


~. 


SIMPLE CURVES. 97 


direction of the radius OJZ/, find on the other curve a point 
having a radius O'# parallel to O/, and measure HK. In 
the diagram draw Od and O'w perpendicular to Hk. Then 
the angle KO'a = 90° — HKU’ = KO'A nearly, which is the 
angle required. We have therefore to find the correction 
a0'A = 2, and apply it to KO'a. 

Aa = f' vers KO'a; Bb = FR vers KO'a nearly. 

Ka = f' sin K0'a; Hb= ksin KO'a 


Bb — Aa = (R—R') vers KO'a | 
ab = HK + (#— Lf’) sin KO'a i | 
(R= R) vers KO'a | 
lig + (R — R') sin HO'a 


sin. @ == nearly. (88) MM 


KO'A = (K0O'a — x) = HOB 


Observe that HO’a = the angle between the tangent at or il 
H and the line 7A; and AKO'A = the angle between the aN 
tangent at 7 or 7 and the required tangent BA. i 

If, instead of H and AH, the points JJ’ and A’ had been Hi 
selected, then vi 
(R — R') vers H'Ob qi 


sin. v= 


and 
H'OB= K'OJA = H'0b-+ 2. 
The length of BA should be obtained by measurement, but HA 
it may be calculated by He 


AB = ab — (R— RB’) sina’ (89) A 


KOA = HOB = KO'a —& 


When R = Rk’, x = 0, and HK is parallel to BA. | ue if 
| | 
In case the curves are reverse to each other, as tn i | f 
Fig. 41, i 
; (R-+ R') vers KO'a HF 
in’ = ——_-___—— —-—— nearly. 90 I 
ane RoR ya KO ae ee ll 


If the points H’ and X’'’ are selected, Fig. 41, 

(R-+ R’) vers H'Ob Ih 
HK —(RtR) aia Hl Ob nearly. (91) it 
H'0B = K'0'A = H'0b +. 


sin? = 


FIELD ENGINEERING. 


The lines HK, AB, and QO’ all intersect in a common 
point J, Fig. 41. 


BELGE OE x 
HI = ELE (92) 
IB = VHI(HI+2R sin HOB) (98) 
iS Sea (94) 


These last three equations furnish another method of 
solving the same problem. They may be applied to Fig. 40 
by changing the sign of R’. 

In Fig. 41, if R= R’, then HT = 44K and AB = 2/8. 


Fia. 42. 


155. Given: two curves, O and O’, reverse to 
each other, joined by a tangent BA', and terminating in 
another tangent, B'F; to change the position of the 
Point of Tangent B of the first curve, so that the second 
curve may terminate in a given parallel tangent, B'F'’. 
Fig. 42. 

Let X be the required new position of B. 

‘* Q" be the corresponding position of 0’. 

te to =A OLE rand peas ARO 

Since the radii and the connecting tangent are unchanged 
in length, and all rotate together about O as a centre, O” will 
be on a circle passing through O', described with a radius 
OO’, and the required angle BOX = 0'00". 


SIMPLE CURVES. 99 


In the diagram, produce 0'A’ and draw the perpendicular 
OG, and let «=the angle 00'G. Also, draw OF parallel 
and O"K and O'H perpendicular to B’O'. In the triangle 
00'G we have 

GO R+&k 


cot OO'G = Goo cota = “eae (95) 
and 
00>-= rel (96) 
COs a 


The angle KOO' = 00'B'’ =a-+ A’. 
The angle KOO" = OO"B" =a-+ A’ 


KO = 00". cos(a + 'A"),, HO = 00’, cos(a« +. A”. 


HK = 00' [cos(a + A")—cos(a + A’)] = BF" 


cos (a -- A"). "cos (@ -- A Se ee (97) 


OX OOO a ee Be) ey (98) 


If we conceive a line to be drawn through O bisecting the 
arc 0'O", the angle it makes with B"O" is a mean between 
B'0'0 and B"0"0; hence the chord 0'O", perpendicular to 
this line, makes an angle with O'P perpendicular to B'O' of 


POO" =4[(@+ 4) ++] 
and since 
O'P = PO" cot PO'O" 


FR’ = B'F' cot t[(«e+t adt(etay] @9 


which gives the distance, measured on the parallel tangent, 
between the old tangent point and the new. 

This problem occurs in practice when both the connecting 
tangent and the radius of the last curve are at their minimum 
limit, and the parallel tangent is énside of the old one, as in 
the figure. Should the new tangent be outside, the same for- 
mulse apply, only changing the sign of B’F’” in eq. (97). But 
in this last case it is usually preferable to employ problem 
§ 136 or § 187. 

Example.—A 1° 40' curve is followed by a tangent of 200 ft., 
and that by a 4° curve of 10 stations ending in a tangent ; 


100 FIELD ENGINEERING. 


and the offset to the given parallel tangent is 80 ft. on the 
inside. Required, the pasition of the new tangent points X 
and B", 

Here & = 3487.87, R' == 1482.69, BA’ = 200, B'F’ = 80. 


Iq. (95) R-+ R’ 4870.56 log 8.687579 


BA! 200. log 2.301030 

“Ge Pa} log cot 1.386549 
Eq. (96) a 2° 21’ log cos 9.999635 
vs 00' 3.687944 
Eq. (97) B’F’ 80 1.903090 
.01641 . 8.115146 


a+ A’ 42° 21’ cos .73904 
a+ A" 40° 56’ cos .75545 


Wy. (98) BOX): 1°25 .-: BR =—s8b5 tt. Ane 
Eq. (99) PO'O" 41° 88’ 80” cot 1.12468 x 80 = 89.97 = #"T' 


156. When the tangents of a proposed road are to be in 
general much longer than the curves, it is desirable to estab- 
lish the tangents first in making the location, and afterwards 
determine suitable curves. On the other hand, if the curves 
necessarily predominate, they should be first selected and 
adjusted to the ground with reference to grade and easy 
alignment, and afterwards joined by tangents. In the latter 
case the field work cannot be successfully accomplished 
unless the location has been previously worked out upon a 
correct map constructed from the preliminary surveys. The 
map Should show contours of the surface, and also the grade 
contour, or intersection of the surface and plane of the grade. 
In side-hill work the grade contour indicates approximately 
the degree and position of the necessary curves. In the work 
of selecting proper curves upon the map, templets or 
pattern curves are almost indispensable. The templets are 
cut to form a series of curves, the radii being taken from 
Table IV. to a scale corresponding to the scale of the map, 
which ranges from 400 to 100 feet per inch, according to the 
difficulty of the location. The templets should represent 
convenient curves, or those in which the number of minutes 


SIMPLE CURVES. 101 


per station bear a simple ratio to 100. Curves of 50’ and 
multiples of 50° are most convenient; 40° curves and multi- 
ples standing next in order, and 30° curves and multiples 
next. 


TABLE OF CONVENIENT CURVES. 


D Ratio of Min. | D \Ratio of Min. D Ratio of Min. 
: to Feet. || : iP to dmeets 1 ; to Feet. 
50’ | a bees 40’ 2:5 30’ 3:10 
1° 40’ 1:1 1° 20’ | 4:5 | 4° QO | 3:5 
2° 30/ | 3:2 R20 00 | 6:5 | 12° 80’ | 9:10 
8° 20’ 2:1 De 40’ 8:5 | 2° 00/ 6:5 
4° 10’ 5:2 |} 3° 207 | 2:1 2° 30/ 3:2 
5° (0! 3:1 4° 0” | 12:5 3° 00/ 9:5 
5° 50’ var) | 4° 40/ 14:5 3° 30’ 21:10 
6° 40’ 4:1 || 5° 207 | 16:5 4° 00’ 12:5 
7° 30 9:2 6° 00’ | 18:5 4° 30/ 27:10 
82 20/ 5:1 6° 40’ | 4:1 5° 00’ 3:1 
g° 10/ 11:2 |! FolOy | . 9222 5 5° 30’ 33 : 10 
10° 00’ 6:1 8° 00/ | 24:5 ‘| 6° 00’ 18:5 


After drawing the curves and tangents upon the map, the 
tangent points and central angles are carefully determined, 
the latter being compared with the lengths of the curves ob- 
tained by a pair of stepping dividers set precisely by scale to 
the length of one station. Field notes are then prepared from 
the map, and if the work has been well done these notes may 
be followed in the field with scarcely any alterations. 

No ordinary protractor will measure the angles closely 
enough for this purpose ; it is better to use a radius as large 
as convenient, of 50 parts. The chord of any arc drawn with 
this radius equals 100 times the sine of one half the angle 
subtended. 

The importance of having absolutely straight-edged rulers 
in such work is obvious. In case a very long line is to be 
projected upon the map, it is well to use a piece of fine 
sewing silk for the purpose. See §§ 53, 54. 


FIELD ENGINEERING. 


CHAPTER VI. 
CoMPOUND CURVES. 
A. Theory. 

157. A compound curve consists of two or more consecu- 
tive circular arcs of different radii, having their centres on 
the same side of tbe curve ; but any two consecutive arcs 
must have a common tangent at their meeting point, or their 
radii at this point must coincide in direction. The meeting 
point is called the point of compound curve, or P.C.C. 
Compound curves are employed to bring the line of the road 
upon more favorable ground than could be done by the use 
of any simple curve. 

When a compound curve of two ares connects two tangent 
lines, the tangent points are at unequal distances from the 
intersection or vertex, the shorter distance being on the line 
which is tangent to the arc of shorter radius. 

158, Let VA, VB (Fig. 48) be any two right lines inter- 
secting at V, and let A be the deflection angle between them. 
Let A and B be the tangent points of a compound curve (VA 
less than VS), and let AP, PB be the two arcs of the curve. 
The centre 0, of the arc AP will be found on AS, drawn per- 
pendicular to VA; the centre QO, of the arc PB will be found 
on BS produced perpendicular to VB; and the angle ASB 
will evidently equal A. Join VS, and on VS as a diameter 
describe a circle; it will pass through the points A and B, 
since the angles VAS, VBS are right angles in a semicircle. 
Draw the chord VQ, bisecting the angle A VB, and join AQ, 
BQ. Then AQ, BY are equal, since they are chords subtend- 
ing the equal angles AVQ, BVQ. From Q as.a centre, and 
with radius QA, describe a circle; it will cut the tangent 
lines at A and B, and also at two other points G and Y, such 
that VG = VA, and VY= VB. Hence BG = AY, and the 
parallel chords AG, BY are perpendicular to VQ. Join AB; 
then AQB = ASB = A, since both angles are subtended by 
the same chord AB. 

In the triangle VAB, the sum of the angles at A and B is 
equal to the exterior angle A between the tangents ; while 
their difference (A — B) is equal to the angle at the centre Q 


COMPOUND CURVES. 103 


subtended by the chord BG, which is the difference of the 
sides (VB —V.A). For the angle VAB = VAG + GAB, and 
the angle VBA = VBY — ABY. But VAG = VBY and 
GAB = ABY, and by subtraction VAB — VBA = 2GAB = 
GQB, since A is on the circumference and Q at the centre. 

159. Turorem—The circle YAGB, whose centre is Q, % 
the locus of the point of compound curve P, whatever be the 
relative lengths of the arcs AP, PB composing the curve. 


Fie. 43. 


On the circle YAGB, and between A and G, take any point 
P, and on AS find a centre O,, from which a circular arc may 
be drawn cutting the circle at A and P; also on BS produced 
find a centre Q., from which a circular are may be drawn 
cutting the circle at B and P. Join PQ, PO, and POs. 
Since when two circles intersect, the angles are equal be- 
tween radii drawn to the points of intersection, QPO:= QAO, 


104 FIELD ENGINEERING. 


and @PO, = QBO,. Draw the chord QS and it subtends the 
equal angles (AO, = QBO.. Hence @PO, = QPO, and the 
radius PO, coincides in direction with the radius PO., which 
is the condition essential to a compound curve. 

Now, if we imagine another point P' to be taken on QP or 
on YP produced, and the arcs AP’ BP', drawn from centres 
found on AS and BS, it is evident that the equality of angles 
found in respect to P could not exist in respect to P’. Hence 
the arcs would intersect in P’ at some angle 0, PO, and would 
not form a compound curve. Therefore, Q. E. D. 

160. THErorem.—Jn any compound curve the radial lines 
passing through the three tangent points A, P, and B are all 
tangent to a cirele having the point Q for its centre, and Jor tts 
diameter the difference of the sides VB and VA. 

Draw the three lines QM, QN, QL perpendicular to the 
radial lines BO,, AS, and PO, respectively. Then the three 
right-angled triangles BQN, PQL, and AQM are equal, since 
BQ = PQ = AQ =radius of the circle AGB, and the angles 
at B, P, and A are equal by the last theorem. Hence QM = 
QL = QN, and if a circle be described with this radius about 
Q, the three lines BO:, PO2, and AO, produced will be tan- 
gent to it. Draw QJ perpendicular to VB; it will bisect the 
chord GB in J; and QV = BI= $BG. Hence the diameter 
2QN = BG = VB —VA; which was to be proved. 

Corollary 1. The compound curve intersects the circle AGB 
in the point P, at an angle equal to half the difference of the 
angles VAB, VBA, For QPL= QBN = BQOT=4BQG. The 
arc AP is exterior, and the arc PB interior to the circle 
AGB. 

Cor. 2. Since both centres are on the line PL, the position 
of the point P fixes the lengths of the radii of a compound 
curve. As P is moved toward G both radii are increased, 
until when P reaches G, AO; becomes AK, a maximum, while 
BO; becomes infinite. As P moves toward A both radii are 
diminished, but the least value of the are AP depends upon 
the least radius allowed on the road. If in the diagram we 
make AQ, equal to the least radius allowed, a right line drawn 
through the point 0, tangent to the circle ZMW fixes the 
corresponding minimum value of the arc AP, and also of 
the radius BO, for given values of VA, VB, and a. Be- 


» | 
COMPOUND CURVES. 105 


tween these limits any desired values of the radii may be em 
ployed. 

Cor, 3. In the triangle SO,0., the sum of the two central 
angles A0,P and PO,B is equal to the exterior angle ASB = 
A; consequently, as the central angle of one arc is increased 
by any change in the position of the point P, the central 
angle of the other will be diminished an equal amount. 

or, 4. Only one value of the angle AO, P is consistent with 
a given value of the radius AQ,, since both depend on the 
variable position of the line PL; and for the same reason only 
one value of the angle BO.P is consistent with a given value 
of the radius BO.. Hence only one radius or one central 
angle can be assumed at pleasure, the remaining parts being 
deducible therefrom in terms of the sides VA, VB, and the 


angle A. 


B. General Equations. 


161. Let 8, =the side VA, S, =the side VB 
Let R, — the radius AN, R, = the radius BO, 
aye dif VAS = BA; A =the sum VAB+ VBA 


“ a, =central angle AOiP, A. = central angle BOP. 
In the triangle BQJ, cot BOL = ae But JQ 2eV i xX 


cot 1QV = 4(S2 +S.) cot dA, and Bl = 4(S_ — S1). 


cot tv = = +5 cot 4A (100) 


a) 
2 


By Cor. 3, AitAr.=A (101) 


In the triangle AQM, AO, = AM— MO,._ But AM = 


MQ cot 4y, and MO, = MQ cot tA. 


Ry = U(S2 — &:) (cot 4v — cot £41) } (102) 


Similarly, Rs = 4(S2—- S:) (cot 4v + cot +A.) \ 
Subtracting, 


R, — Bi = (Sa — S1) (Cot 4A2 + cot $A1) (108) 


FIELD ENGINEERING. 


| RAL aa ora | 
plO2 — Al 
From (102), (104, 
RR 
cob PAs = 7g gy cota | 
In the triangle ABG, 
AB sin BAG 
coats sin AG@V 
or 
3AB sin 4 
tAiseiQdiceP ae (105) 


sin tA 


by which we find 3(S, — S,), when, instead of the sides and 
A, we have given AB, and the angles VAB and VBA. 


Rea 
From (103), 4(S: — 8,) = a roe meee TONE (106) 


R 
From (102), (107) 
Re 


cot $v = —.— —; — cotia 
bi EL ESS Hi 


3(S2 — S,) cot 4 
From (100) 4/8, + &) = 28s aa By (108) 


S, and S, are found by adding and subtracting the values 
found by egs. (106), (108). 


$(S_ — 8) sin 2A 
sin dy 


From (105), 44B= (109) 


ti which may be used instead of (108) when the sides are not re- 
| quired. VAB=}(A+y)and VBA=i(a — y). 

162. Given: the sides VA = 8, and VB = &, and the 
angle A; assuming the shorter radius R,, to find Ai, Aa, 
and Ro. 

Use equations (100), (104), (101), (102), and (18). 

Hzample.—Let VA = 1899.90, VB = 1091.12. a = 74°, and 
assume A, = 955.37. 


COMPOUND CURVES. 107 


(100) 4(S2 + S;) 1495.51 
1(S, — S,) 404.39 


LA Hf 


a a 
04), G, (Uv = 6) 


(Sa — 81) 
; pA 21° 27’ 
(101) 4 tA Sis 
wags Aa 15° 33’ 
(102) 37 


$(S, — S1) 
R, (D = 1° 40’) 


log 3.174789 
2.606800 


< 0.567989 
cot ‘* 0.122886 


11° 31' 01".5 cot 4.90769 “ “ 0.690875 


2.980170 
‘** 2.606800 


2.36249 ‘© 0.378370 
cot 2. 54520 


3.59370 
“« 4.90769 


8. 50139 «0.929490 


2.606800 
3 536290 


(18) Prd tay SS eee 42° 54’, Dy = ilo No 31¢ 06’, Dg — 1866. 


163. Given: the line AB, and the angles VAB, VBA; 
assuming the longer radius R., to tind A>», Ss, and Ry. 


Example.—Let AB = 2487.82, VAB = 48° 31, VBA = 25° 29, 


and assume R, = 3437.87. 


(105) AB 1218.91 


iy 11° 81’ 
i 37° 
- (4 — 81) 404.88 
(104) Re 
Wy 11° 81! 
ie Seas a 15° 33' 
101) $A oh 
ey 21° 27 
i102) 4y 
3(82 — 81) 


log 8.085972 
sin “ 9.300276 


‘2.386248 
ce 6 9.779468 


«© 9606785 
3.536289 


8.50166 ‘< 0.929504 
cot 4.90785 


cot 3. 59381 


cot 2.54516 
«© 4.90785 


9. 36269 log 0.373407 
2.606785 


2.980192 


108 FIELD ENGINEERING. 


164. Usually a compound curve is fitted by trial to the 
Shape of the ground, after which it may be desirable to 
ealculate the sides VA, VB, or the line AB, and the angles 
VAB, VBA. 


Example.—From the point of curve A, a 6° curve is run 
715 feet to the P.C.C.; thence a 1° 40’ curve is run 1866 feet 
to the P.7. Required, the sides VA, VB, and the line AB, 
and angles VAB, VBA. Here R, = 955.87, A, = 42° 54, 
fi, = 8487.87, A = 31° 06’. 


(106) R.— Rh, 2482.50 log 3.894889 


$Ai 21° 27 cot 2.54516 

$A2 15° 33! ** 3.59370 
6.13886 ** 0.788088 
. 4S. — S,) 404.39 «2.606801 
(107) A, «2.980170 
2.36248 ‘< 0.878369 

tar 21° 27’ cot 2.54516 


ey, 11° 81' 01.7 “4.90764  “ 0.690873 
(108) $(S2 — S;) 2.606801 


‘* 3.297674 


2A 37° cot “ 0.122886 
4(Sa + S;) 1495.51 “3.174788 
Ss 1899.90 
8, 1091.12 
VAB 48° 81’ 
VBA 25° 29' 
(109) 4(S2 — S,) “2.606801 
tA 87° sin “ 9.779463 
2, 386264 
ly 11° 81’ 01".7 sin “ 9.300294 
1AB 1218.91 3.085970 
AB 2437, 82 


165. Given: the radii R,, Ra, the angle A, and one xde, 
VA, or VB, to find the other side and the central angles Ay, 


COMPOUND CURVES. 109 


In the triangle AM/Q, AO, = AM — MO; = IQ — MQ cot 
MO,Q: or 


R, = 4(S2 + Si) cot $A — $(S2 —Si) cot FA1 
whence 


4(S, +. 8) = 4(Sq —S,).cot tA) tan 4A + R, tan 4A 


By eq. (106) 
sin +Ae sin $A 
sin 4A 


4(Sq — 1) = (Re > R;) 
Substituting this above, subtracting and reducing 
Sy a dais — Ry) sin +Ag pais AS ot + R, tan 4A 


But #(A — A:) = $A, and 2 sin? $A. = vers Ae, whence 


i= (R. — fy) vers A: + Ri vers A (110) 
sin A 
Transposing, Ht 
S, sin A — fi, vers A a 
vers Ace = a eee (111) | 
Similarly, from the triangle BQO: 7 ad 
1) 


it —— 4(S, = S;) cot tA ob 4(S2 aad 81) cot +Ag i) H | 


from which and eq. (106) we derive 


2, vers A — (RR. — Ay) vers A | 
g, Ba vors A’ (Ra— Ri) vers Av aia) | 
sin A | 
and 
R, vets A — S. sin A (112) 


ai 3 i 


119 FIELD ENGINEERING. 


Hvample.—Given : VA = 8, = 1091.12, A = 74’, and the 
radii R, = 955.37, Rz = 3437.87, to find A,, As, and S. 


(111) S, 1091.12 log 3.037873 
A "4° sin ** 9.982842 

1048.85 “ 3.020715 

Ri © 2.980170 

A 74° vers *£ 9.859956 

692.03 “9 840196 

! 306.82 «© 2.552449 
i R, — R, 2482.50 ‘3 394889 
Hi Ke 81° 06’ vers “ 9.157566 
1 ASIN 42° 54! «46 9 497954 
1 (112) R, — R, “3 394889 
663.96 “9999143 

R, ‘© 3536289 

A vers ‘* 9.859956 

2490.26 “3 396245 

1826.30 “3 961572 

A sin “ 9.982849 

Sa 1899.90 “3 978730 


166. Given : one side, and the radius and central angle of 
the adjacent arc, to find the other radius and side. 
From eqs. (111), (118) we have 


S, sin A — R&, vers j 

1 |. y ers A 

Eg Te Se 3 Se sat - 
vers Ag 


(114) 
ey vers A — So sin A | 
vers Ay 


Be Ri = 


by one of which the required radius may be found; the required 
side is then found by eq. (110) or (112), as in the last problem. 

Lzample.—Given : VA =S; = 1091.12 a = 74°, R, = 955.37 
and A; = 42°54’; to find R, Aa == 74° — 42° 64’ = $1° 06 


COMPOUND CURVES. Lil 


(114) ref 1091.12 log 8.037873 

K "4° gin 9.982842 

1048.85 << 3.020715 

R 955.37 | “ 9.980170 

A 74° “ vers 9.859956 

692.03 “ 2.840126 

356.82 “ 9.552449 

a 31°06 © vers 9.157556 

a. Ry — R, 2482.52 «“ 3.394893 
SoBe 3437.89 


Fig. 44, 


Otherwise: Fig. 44. If convenient in the field, a tan- 
gent PV, may be run from the point P to intersect the 
farther tangent. The -distance PV, multiplied by cot $4; 
will equal the radius R, by eq. (25). 


167. Remarks.—If the first arc AP be produced to G, 
Fig. 44, so that A0,@ = A, then Gis the tangent point of a 
tangent parallel to VB, and by $187, the tangent point B must 
be on the line PG produced. Conversely, if the point B is 
assumed, and the arc AG given, the point P must be on 
the line BG@ produced. The radius R, may be found by 


112 FIELD ENGINEERING. 


sho LP being measured on the ground ; 
similar triangles R,: R, :: BP : GP. 

The distance VD, Fig. 48, from the vertex to the circle 
AGS is expressed by the formuia 


ite 


or by 


ex'sec ey 
sin dy 


VD = 3(82—8)) - 


(115) 


Tf the point P falls at D, then VD is also the distance of the 
curve from the vertex measured on the line VQ. But when 
P falls at D, the radius PO, is perpendicular to the line AB, 
and A, = VAB, and A, = VBA. When a, is greater than 
VAB, the arc AP, being exterior to the circle, cuts the line 
VD; but when A, is less than VAB, the arc PB cuts the line 
DQ. 

If the line O.P produced passes through V, we have 
S.— 
i+ & 
giving A: = 4A + QVLand As = $A — QVL. 


When A, is greater than this, we have for the external 
distance of the vertex 


SiR Wy Da ts Sin (116) 


Hi, = R, ex sec AO, V 


in which the angle AQ, V is found by the formula cot AO, V= 


de, 
oi? and #, is measured on a line VO,, making the angle 
£51 


AVVO; = 90° — AO,V. 
When A, is less than (}A + QVL), we have similar expres- 
sions with respect to the arc BP and centre Oy. 


168. To locate a compound curve » when the point of com- 
pound curve is inaccessible. Fig. 45. 

Hach are being in itself a simple curve is located as such. 
When the P.C.C. is accessible, the transit is placed over it, 
and the direction of the common tangent found, from which 
the second arc is then located. 

When the P.C.C. is not accessible, the common tangent 
Vi V2 may be found by locating the points V,; and V2, which 
may be easily done, since V:A = ViP= R, tan +Aj, and 


COMPOUND CURVES. 113 


V.B = V2P = ft, tan 4 Ae, from which eacl: are may then 
be located by offsets or otherwise, as in the case of simple 
curves 

Should the points V,; V2 be obstructed, the common tangent 
may be found by an offset 7G = LP from any convenient 
point /7, for knowing the angle HO,P, we have HG = R, 
vers 110, P, and. GP.= & sin -HO,P. 

If the cutire tangent V, V.is too much obstructed for use, 


the parallel line ZA may be employed, observing that the 
Ae -gee ey weatdil. 
angle PO. is found by vers PO.K = Re and the distance 
v2 
LE by LK = R, sin PO,K, by which a point # on the second 
arc is found having a tangent offset KAJ = HG. 


Fig. 46. 


Fic. 45. 


Should the line HK be also obstructed, we may run the in- 
verted curve HP’ = HP and P'K = PK to find the point K 
from which so much of the second are as is accessible may be 
located. 


C. Special Problems in Compound Curves. 


169. Given: a compound curve ending in a tangent; to 
change the P.C.C. so that the curve may end in a@ given 
parallel tangent. Fig. 46. 

Let APB be the given curve ending in VB, 

«< V'B' be the given parallel tangent, 
‘‘ » = perpendicular distance between tangents. 

It is required to change the point P, and with it the values 
of A, and Ag, so that with the same radii R, and AR, the new 
curve AP’B’ may end in the parallel tangent V'B. 


114 FIELD ENGINEERING. 


a. When the tangent V'B' is inside of VB: 

Let A, = AGP, Ay = AOLP’, Ar= PO.B, Ad’ = P'0,8B, 
and in the diagram draw 0G perpendicular to BO,; then 
GO, = 0:0, cos Ao, KO, = 0,0. cos Ao’. Subtracting, 
since 0:0. = 0,02' = (Rh, — fa), and K0O;' ~ G0, = GB— 
AB =p, 

p= (hk; — A) (cos Ay’ — COs Ae) 
whence 
pe eee Ay (117) 


cos Ai = 
dis ee R, : 


PO,P' =(A2 — Ad’) and the point P is zdvanced, 


b. When the tangent V'B' is outside of VB: 
| p=(R. — R,) (cos Aa — COS Az’) 
i i whence 


ee S P 


PO, P’ =(Ae’ — As)and the point P is moved back and the 
arc AP diminished. 


Fie. 47. 


| In case the curve terminates with the arc of shorter 
‘| radius, or R, follows R.. Fig. 47%. 
ec. When V'B is inside of VB: 
p= (R = Ri) (cos Ay — COs Ai) 
whence 


cos A, = cos Ai — Peas (119) 


POP’ = (Ay — «A1) and the point P is moved back. 


COMPOUND CURVES. 115 


d. When V'B' is outside of VB: 
p = (fig — Ri) (cos Ay’ — cos A:) 
whence 
p 


0S Ai = Os Ai Fiols B (120) 
bg — 401 


PO,P’ =(A: — 41) and the point P is advanced. 
Example.—Let R = 9992.01, Ri = 1432.69, Ae= 28°, and 
p = 20.07 inside of VB; case a. 


p 20.07 log 1.302547 
(117) R, — R, 859.32 ‘* 2.934155 
023356 ‘* 8.368392 
As 28° cos .88205_ 
A's 25° ‘* 906306 
POG: 3° 


170. Given: a compound curve terminating mn a tangent, 
to change the P.C.C. and also the last radius, so that the 
curve shall end in a parallel tangent at a point on the 
same radial line as before. Fig. 48. 


Fig, 48. 


Let APB be the given curve ending in the tangent VB; let 
V'B' be the given parallel tangent; and let 9 = BBA= Hl= 
tne perpendicular distance between tangents. 

' Tt is required to change the point P to P’, and also the 
value of R, to R,', so that the new curve May endin V'B’' at 
2’ inside of VB on the same radial line BOs. 

In the diagram produce the ‘are AP. to G to meet 0:4 
drawn parallel to 0.B; then PO,G = Az. Draw the chord 
PB, and it will pass through G. Lay off the distance p from 


116 FIELD ENGINEERING. 


Bon BO; to find B'; draw B'G and produce it to intersect 
the arc APG in P'.. Then P’ is the P.C. C. required. Join 
P' 0; and produce it to meet BO: produced in 0... Then 
P'0.' = B'0,' = Rf,’ the new radius, with which describe the 
are P'B'. 

By Geom. Tab. I. 18: 

PBV =} P0:B = 44a, and @B'V' =4P'O.B =4A0'. 

PGP’=BGB' = #A2— A?) 

Draw 0,K perpendicular to BO. 

Then OK = BH=s B= 0; O. sin Gy = (Re == R) sin Ag 
GH 2 GI s2 x 


GI 
tan tO. = pr tan Ae SBC) EBLE 


p 
t tAo == te 4 — —— F 
antAce ané+Ae (Racadhisin Ds (121) 


In the triangle 0:0:02' 
sin Ae’: in Aat: 0100: O100'::: (Re. —)Ra) 3\(Re = fi) 


and ‘ 
mae ME |< cla. 
Ry == (be Ry) Tae: + fy (122) 


If B'V' were outside of VB; 


p 

= tan ? : 9 

tan Ao an 4Aqgt+ (R; Se R,) sin A‘ (1 3) 
Ri = (Rae hee (122) 


sin Ag 


When the smaller radius R, follows R,: If the given 
tangent B'V' is inside of BV. Fig. 49. 


tan 4A,’ = tan4Ai + Aiea . (124) 
1 1 
j Mpg eT PARIS, ca gee (125) 


sin Ai 


COMPOUND CURVES. 


If B'V' is outside of BV: 


pid jhe ye “OS eee 
tan Ai SS valk 3.28 1 (R; — R) bins Aa (126) 


Rico haa. RD So (125) 


Fia. 49. 


Example 1.—Ig. 48. 


Let R, = 2292.01 p = 20.07 inside. 
“« Ry, = 1482.69 Ag = 28° 


(121) Ry — R, = 859.32 log 2.984155 Mh 
As 28° log sin 9.671609 Hh: 
2.605764 | 
p 20,07 1.302547 | 
04975 8.696783 IM 
tan Ae 24933 | 
.*. tan $Ae! 19958 ab atg Wh 
Gea 92° 84’ sin 9.584058 Vn 
(Ry B,) 2.934155 i 
3.350097 | 
i 28° sin 9.671609 
(Ra’ ~ Ri) 1051.25 3.021706 
R, 1482.69 


Ans. R,! 9483.94 .*. D = 2° 18' 26" 
PO P == 128° 22°84) = 5° 26°. PPo = 135.83 ft. 


1is 


FIELD 


Keample 2.—Fig. 49. 


Let R, = 2292.01 pp 


ENGINEERING. 


= 20.07 inside. 
— AG. 


(124) R,— R, 859.82 log 2.934155 
Ai 46° log sin 9.856934 
2,.'791089 
p 20.07 1.302547 
08247 8.511458 
tanyA, 42447 23° 
tan $A,' 45694 24° 333! 
SNE 49° 07’ log sin 9.878547 
Peer 9.934155 
3.055608 
Nie 46° log sin 9.856934 
817.60 2.912542 
Re 2292.01 
Ans. »| Ry 1474.41 0D =B° 53" 12" 
PO:P! = Ay! — A: = 8°07.” are PP’ =" = 124.67 ft. 


Observe that in either figure both tangents must be on the 
same side of the point G, in order to a solution. 


Fra. 50. 


17. Given: a compound curve ending in a tangent, to 
change the last radius and also the position of the P.C.C., 


so that the curve may end in the same tangent. 


Fig. 50. 


y 


COMPOUND CURVES. 119 


I. When the curve ends with the greater radius Re. 
Let APB be the compound curve in which R; R, A, and 
A» are known. 
in the diagram draw the chord PB and produce the first 
arc AP to meet it in G; draw 0,G, and produce it to meet the 
tangent in K. Then by § 137 O,K is parallel to 0.B, and by 
eq. (57) MI 
GK = (R. — RB.) vers Aa (127) ni 


If we assume P' as the new P.O.C,, we have A2'= P'0,'B,, 
and the chord P’G produced will intersect the tangent at the 
new point of tangent B’, and BO,' = R,'. . Similar to eq. (12) | \ 
we have HH 
GK = (R;' — R,) vers A: Wi 


and equating the two expressions, we obtain HA) 


rae (R, — Ri) vers Az _ GK 
Fy! = Fa vers Ae pie vers Ad eo) He 


If we assume Raz, we have vee 


R, — hy GK 
Fil hts Cate a ere A 2 
vers Aa = By Bye pied ah Bas Bh , 


In the two right-angled triangles BAG and B'KG, we have 


BK = GK cot tAs i 
BK = GK cot 4A: Win 


and by subtraction, | 
| | 
BB’ = GK (cot 443' — cot 44s) (130) | 
in which GA is obtained from eq. (127). ti} 
When BB’ as given by eq. (180) is negative, the point B’ falls He 
between B and V. 
If we assume the distance BB’ on the tangent, we have 
from the last equation, 
BB' 


cot Aq — cot +Ae 3 GK (131) 


120 FIELD ENGINEERING, 


GK being obtained from eq. (127) and FR,’ from eq. (128). In 
eq. (181) use the + sign when B' is beyond B as in the Fig. 50. ' 

Il. When the given curve ends with the smaller radius 
R,. Fig. 51. 


Fig. 51. 


We have by a similar reasoning 


GK = (R, — Ri) vers A, (132) 
2 — Ry 1 
vers A, vers Ai 
1 fe fy ~__ GK 
vers Ai = era, vers: Ay = Drie he (134) 
BB = GK (cot $41 — cot $A1) (185) 
BRB 


cot¢A, = cot pA, += (136) 


GK 
using the — sign when £' is beyond B. 
Hxanvple.—Fig. 51. 
Let R. = 2292.01, RA, = 1482.69, aA; = 46°, and let the 


P.C.C. be moved back 200 feet from P to P’; hence PO,’P’ = 
5° and A,’ = 51°: to find the new radius R,' and the distance 


4) 


COMPOUND CURVES. 


log 2.934155 


Kg. (182) R2 — BR, 859.32 
Ai 46° « vers 9.484786 
7. Ge log 2.418941 
eq. (188) Ai’ 51° « vers 9.568999 
R, — Ri 707.85 2.849942 Hy 
Ry 2292.01 il 
syn 1584.16 and D = 3° 87’ ) 
eq. (185) GK log 2.418941 Hine 
cot $A, 2.35585 23° A 
cot $A1' 2.09654 25° 30' | A 
0.25981 log «9.418819 it 
- BB 68.04 1.832760 i 
172. Given: a compound curve ending in a tangent, the last | i} 
radius being the greater, to change the last radius and AW 
also the position of the P.C.C. so that the curve may end at the 1 


same tangent point, dui with a given difference in the 
direction of the tangent. Fig. 92. 


Let APB be the given compound curve, PO, = R, and 
PO, = R, > fi. Bi 
Let V’Bbe the new tangent, and the angle VBV' =, the i I 
given difference in direction: to find BO,’ = R,', BO,’P' = H 
A,’ and the angle P0,P'. 


g 


{22 FIELD: ENGINEERING. 


We have 
Bo, oe 0,02 = Re Fi (Re <= R,) = Ri 
Bo,’ a 0, 0, = R, a (Re'— Ry) = Ti; 


From which we see that whatever may be the value of the 
new radius, the difference of the distances from B and O, to 
the new centre is constant, and equal to Ri. We therefore 
conclude that the centres 0, and OQ,’ are on an hyperbola of 
which B and O, are the foci, and R, the major axis. 

This suggests an easy graphical method of solving the 
problem. 
tilt Through B draw a line perpendicular to the new tangent 
At V'B which will give the direction of the required centre O,', 
He On this line lay off BK equal to A, and since (y’ — hy) = 
a 0,0.' = K0,', if we join KOQ,, the triangle KO,'0, is isosceles; 
i} therefore bisect KO, and erect a perpendicular from the mid- 
dle point to intersect the line BA produced in 0,'. Draw 0,0, 
and produce it to intersect the are AP (produced if necessary) 
in P'.. Then P’ is the new P.O.C. required, and BO,’ = 
P'0O,' = R,', the new radius. 


The analytical solution 1s as follows: 
Adopting the usual notation of the hyperbola 


Let 22 = R, = the-fajor axis, 
‘© 9 — BO, = the distance between foci. 


Produce the arc AP and through B draw the’ tangent BH, 
and join HO, = R,. Then im the right-angled triangle BHO, | 


1) BH? = BO, — R,? = 4c° — 4@? 
Now by Anal. Geom., c? —a@’ = 0’. 
Therefore 2b = BH = the minor axis. 
Draw the chord PB and produce the arc AP to cut it in G 
Then by Geom. (Table I. 24) 
BH*?= PB x GB = 2R, sin +Ag x 2(Re — R,) sin +Ao 


= 2 sin dae Ratha — Ba) (137) 


ny 


COMPOUND CURVES. 123 


Let a = the angle HO,S, then 


BH R, 


tan Q = Pie and Bo, =. Coe a (138) 
In the triangle BO, 02 let 0,BO. = f ; then 
igi Ras Bi | i} 
vg eh Bo, sin Ae (139) iM 


The polar equation of the hyperbola for the branch 10,02’, 
taking the pole at B and estimating the variable angle » from 
the line BO,, is Hy 
c¢.cosv—a ii 


r= 


Wheno =f +4, r= Ry, and substituting the values ot 
a, b, and ¢ found above, we have 


BH? i 
2 (BO, cos (6 + 7) — fi) (140) at 


Ry — 


using (+7) when V’ falls between V and A, as in the 
figure, and ( — 7) when V’ falls beyond V. 
In the triangle BO,0,', the angle BO.'O, = As’ and 


Ha 
Bo, } Bi & 


sin As’ = a7 qe sin (f + t) (141) We 
Finally 
EE STOR 6 RIE ae | 


Remark.—When V’ falls between V and A, as in Fig. 52, if | 
the angle é be greater than the angle VBH, the curve ceases to 1 
be a compound, and becomes reversed. Therefore VBH = | 
«a — fis the maximum value of 7 possible in this case. When , id 
V' falls beyond V, the point P’ will fall between Pand A; ll 
and the largest possible value of 4 will then be that which 
renders PO, P’ = Ai, and makes the point P’ coincide with A. 


124. FIELD ENGINEERING. 


Example.—Fig. 52. Let Ri = 1482.69 Ai = 31° 
a=. = 2292.01 As = 56° 


2 


(137) R, — RB, 859.82 log 2.984155 
R, 2292.01 3.360217 
2) 6.294372 
3.147186 
1KY 28° log sin 9.671609 
2 0.301030 
a BH 3.119825 
(138) R, 1432.69 3.156151 
a 42° 36' 28".7 log tan 9.968674 
a 43°. 36) 23.7 log cos 9.866889 
os, BO, 3.289262 
(139) R, — R, 3 2.934153 
9.644893 
Aa 56° log sin 9.91857 
B 21° 28’ 06".3 log sin 9.563467 
(140) B+a 27° 98’ 06".3 log cos 9.948053 
BO, 3.289262 
1727.09 3.237315 
R, 1432.69 
294.40 X 2 = 588.80 2.769968 
BE? 6.239650 
Ry’ 2949.05 3.469682 
(141) .°. Aa’ = 36° 18’ 26" 
(142) .°. PO,P' = 18° 41' 84" = 342.8 feet. 


Remark—This problem may also be solved by first finding 
the new sides V'A, V'B, from which and the new central 
angle (A + 7), and the radius #,, may be found A,’, Az, and 

2.’, as in $162. The new sides are readily found from the 
old ones by solving the triangle VBV’. If the original sides 
are not given, they must be calculated as in § 164. 


173. Given: a compound curve ending in a tangent, the 
last radius being the less, to change the last radius and the 
position of the P.C.C. 80 that the curve may end at the same 
tangent point, dut with w given difference im the 
direction of tangent. Fig. 53. 


ny 


COMPOUND CURVES. 125 


Let APB be the given curve, and PQ: = R2,and PO; = 
R, < R. Let V’Bbe the new tangent, and VBV' =1, the 
ziven angle; to find BO,’ = R,', BO,'P'= Ay, and PO.P’. 

We have 

Bo, + 0,02 — Ri; + (Re — RF) = R, 
Bo,’ +. O05 a RS + (R, ae Rs) => R, 


from which we infer that the locus of the centre O,' is an iN 
ellipse, of which B and OQ, are the foci, and & the major axis, . 


Fie. 53. 


since the sum of the distances BO,’ and 0,0,’ is always equal Bi 
to Re. 

This suggests an easy graphical solution of the prob- 
blem, as follows : 

Perpendicular to V'B draw the indefinite line BA, which Hy 
will contain the required centre 0,', and lay off BK = fy. | 
Join KOxz, bisect it, and from the middle point erect a perpen- Hi | 
dicular to intersect BK in 0,'.. Join 020,’ and produce the Hi 
line to intersect the arc AP (produced if necessary) in P’, | 
which is the new P.0.0. required. P'O.' = BOS Br abhe 
required radius, and P! Of Bid 1: 

The analytical solution is as follows: Adopting the 
usual notation of the ellipse, 


let 24 = R, = the major axis, 
‘© 9¢ — BO, — the distance between foci. 


At Berect BH perpendicular to BO; to intersect the arc AP 


w 


126 FIELD ENGINEERING. 


(produced if necessary) in H, and join HO, = R,. Then 
BH? = R,? — BO,’ = 4a — 4¢ 

But by Anal. Geom., a? —. ¢? = 0. 

Hence 2b = BH = the minor axis. 


In the triangle BO,O, we know BO, = R,, and 0,0; = 
R, — R,, and the included angle BO,O0, = 180° — A,; hence 
by Trig. (Tab. II. 25) 

tan 3(0,0.B — 0,BO:2) = a tandAi (148) 
2 
Wy The angles at B and 0, are then found by (Tab. II. 26). 
Ail Let 6 = the angle 0,B0.; then 


| sin A, 


| BOs = (ha — fis) sin 6 (144) 
The value of BH? above may be written 
BH? = (R, + BO) (R2*— BO2) (145) 


The polar equation of the ellipse, taking the pole at B, and 
estimating the variable angle v from the axis BOs, is 


}? 
~ &@—C. COSY 
When » = 6 ¥ 7, then r = Ry’, and substituting the values 
of a, b, and ¢, given above, we have 


BH? 


th = 9(R, — BO, cos (B ¥ 4) (146) 


HA using (# —7) when V" falls between Vand A, as in Fig. 93, 
ni and (8+ ¢) when V’ falls beyond V. 

| in the triangle BO,'O,, the angle O,'BO, = (fF ¢), and the 
exterior angle BO;'P’ = Ax‘; hence 


BO ; ; 
oa Pe sin (f F 7) (147) 


sin Aa = 


Finally PO,P' =(A, #¢)— Al (148) 


When V’ is on AV, then POP’ is negative, showing that 
it must be laid off from P toward A; but when V’ is beyond 


COMPOUND CURVES. 127 


V, then PO,P’ is positive, and P’' will be on AP produced 
The only limits imposed on the angle ¢ are that the resulting 
value of PP’ shall not exceed PA, and that R,' shall not be 
less than a practical minimum. 


Example.— Fig. 538. 
Let Dy =.3° 20’. Ra=.1719.12. Ar = 28° 20° 


ip Us R, = 955.87 A, = 48 t= 45 

The resulting values are as follows: 
p 21° 09’ 82".6 

BO, 1572.42 3.196567 

HH? 5.688829 

Ry 1273.65 3.105652 
Ay 54° 56 
POP! 14° 41’ 


2 
PE, 440.5 
(See also remark at end of $172.) 
174: Given a simple curve joining two tangents, to re- 


place it by a three-centred compound curve between 
the same tangent points. Fig. 54. 


Fig. 54. 


Let R = AO = radius of simple curve. 
R, = P0O,=P'0A,<h A, = PO,P' 
R, == AO, aa BO; — R INCU = AC a BO;P' 
AOL 
Since AQ, is made equal to BO; and VA = VB , AO.P musi 
equal BO,P', and the compound curve will be symmetrical 
about the bisecting line VO; and the centre O, will be on the 
line VO. : 
We have at once from the figure, 
2h 4- ets (149) 


128 FIELD ENGINEERING. 


In the triangle 00; 02 we have 
0:0.: O02 :: sin AOV: sin POV 
whence 
R, — R)sin $A 
(By S'B)aingA a0 
sin 3 Ay’ 3 

which expresses the general relation between the quantities, 
Rand A being given. 

We may now assume values for Ry and Res subject to the 
above conditions, viz., Ri < Rand Rk, > Ah; whence 


sin +A — Sa (151) 


In selecting values for A, and fs, the degree of curve D, 
should be but little greater than D of the simple curve, say 
from 30 to 60 minutes, while Dz may be taken at 4D to 4D. 


Eeample.—Given: R= 1719.12 D= 3°. 20'. A = 40° 


Let R, = 1432.69 D, = 4° 

«© R, = 5729.65 D, =1° 
Ry — R 4010.53 
R, — R, 4296.96 


log 3.603202 
3 633161 


“« 9.970041 
tA . 20° log sin 9.584052 
tA 18° 36°57" ‘* “ 9.504093 
A 87° 13' 54" 
Ae 1° 23° 038" 
AP = P’B : 138.48. 
Again we may assume A, and fy, whence 
AN er Aa 7 2Ae 
and 
. #sinta — R, sinéAy ve 
Fees Kinin Gn, <Sstiag Ai eu 
Example.—Given. R= 1719.12 A= 40° 
Let R, = 1482.69 A,=1° .*. Ai = 38° 
Ans. Ry = 7887.24 «+. Dy = 0° 464° AP = 129. 


Finally we may assume Ay and feo, and deduce A, and &; 
from eqs. (149) (150); but this is the least desirable because 


ay 


COMPOUND CURVES. 129 
the value of R, so found will not usually give a convenient 
value to the degree of curve D. 


175. To determine the distance HH’ between the middle 
points of a simple curve and a three-centred compound curve 
joining the same tangent points AB. Fig. 54. 

In the triangle 00:02, we have 


(Rp, Rye Aa 
O00, os (R, R,) sin $A 
HH' = 00, + 0,H' — 0H | 
HHH = (Bs SR) — (R— By *(158) 


sindA 


in the first example given above HH’ = 14.55, and in the | 


second HH' = 17.05 ft. 

In many instances the distance HH’ is so great as to render 
this problem practically useless, unless the distance HH, is 
discounted beforehand by putting the simple curve AHB a 
sufficient distance inside of the proper location through the 


point H'. But the problem given below is usually preferable. Wi 


176. Given, 4 simple curve joining two tangents to re- Vee 
place it by a three-centred compound curve which Ha 


shall pass through the same middle point H. 
I. The curve flattened at the tangents. Fig. 55. 


Let R= AO, the radius, and A = the central angle of the 
simple curve AHB, and let H be the middle point. 


Midas Ope. 
Decade Oa. maka GED. 


Let Ti = PO; — HO, 
Es Fo = BOs = A' 0. — B'O0s 


« A’ and B' be the new tangent points required. 
We have at once, as in the last problem, 


2ZAe+ Ai= A. 


130 FIELD ENGINEERING. 


Since the curve is to be symmetrical about VO, BP = RS 
PA = P'B, and AA' = BD’. 


Bt 


In the diagram produce the arc HP to G, and draw OG 
parallel to OA, and produce it to K. Then a tangent line al 
G will be parallel to VA; and by § 187 the point G will be ov 
the long chord HA, and on the long chord PA’. GK is the 
perpendicular distance between parallel tangents, and the 
problem is similar to that given in $171; whence by eq. (57) 
we have, in this case, 


GK = (R, — R,) vers Aa = (R — Ri) vers 4A. (159) 


for the general equation in which & and A are given, 
Analagous to eq. (180) we have 


AA' = KA' — KA = GK cot GA'K — GK cot GAK. 
- AA' = GK (cot 4A2 — cot $A) (156) 
in which GK is obtained from (159). 
We may now assume values for R, and Rz, making RB, < R 


and R, > R, and deduce the values of As, Ai, and AA’, 
Solving eq. (155) 


_ (R— Ry) versta _ GK 
Were Ay Cr a ee nly (157) 


Eq. (154) gives A., and eq. (156) gives AA’ 


COMPOUND CURVES. 


Example.—Fig. 55. 


Given: R= 764.489 D = TV 30’ 

Let R, = 716.779 ye hl 
“« R, = 3487.870 Dz = 1° 40’ 
(155) R — Ay 47.71 
+A 20° 


GK 
R,— Ry 2721.091 


Ae (say) 2° 38° 
ie A'P 158.00 Ai = 34°44 
(156) tAs 43.5081 = cot 1°19 


+A 5.67138 cot 10° 
37.8368 

Gk 

AA’ 108.87 


131 


A = 40° 


log 1.678609 
log vers 8.780370 


log 0.458979 
° 3 434743 


log vers 7.024236 


log 1.577914 
0.458979 


«© 2.036893 


Again, we may assume Az and Ay < R; whence 


A, = A — 2A: 


and 
eq. (155) GK = (Rk — &) vers tA 
and 
GK 
fis = Briere Ae 


Eq. (156) gives AA’. 


(158) 


Again, we may assume Az and the distance AA’, whence, 


from eq. (156) . 
AA' 


GK = : 
cot $A2e— cotzA 


GK 


AACE = R — —_— 
eq. (155) R= R SAN 


eq. (158) gives Re. 


(159) 


Again, we may assume R, < Rand AA’; then, eq. (155) 


= (R — R,) verstA 
and eq. (156) 


cot 4A, = cotta + = 


and eq. (158) gives Rs. 


(1603 


132 


FIELD ENGINEERING. 


Hxample.—Fig 59. 


Given: R = 764.489 DZ T 3 A = 40° 
Let) Ah, = 716,779 BD, = 
ee AA: 
Hence by last example, 
GK log 0.458979 
eq. (160) AA’ 110. 2.041393 
38.2309 1.582414 _ 
cot $A 5.6713 10° 
cot tAe 43.9022 1° 18’ 18” log 1.642486 
(158) Ae (say) 2° 87 log vers 7.018147 
GK 0.458979 
R,— Ri = 2759.5 3.440832 
R, 3476.3 Deg == lod 
AP’ 157. Ai = 384° 46 


Il. The curve Sharpened at the tangents. Fig. 56. 

This case will only occur when, with a given external dis 
tance VH, a simple curve would absorb too much of the tan- 
gents. 


Fie, 56. 


Let AHB be the simple curve, and 
“ 4'PHP'B'the required compound curve, 
hg == PO. = HOF A, POF 
“ R= PO, = AO. B'0,; 4:2 A OP= P'0;B', 
We have from the figure, 
24Ai+ Bo= &. (161) 


COMPOUND CURVES. 139 


In the diagram draw 024 parallel to OA cutting the tan- 
gent at K, and produce the arc HP to G. Draw the chords 
GH and GP, passing through A und A’ respectively. We 
have then a discussion similar to the preceding case, and to 
the problem § 171, Fig. 51, whence we derive the general 
formule: 

GK = (R, — Rh) vers Ai = (R. — R) vers +A (162) 
and 
AA' = GK (cot $A: — cot ¢A) (163) 
1. Assuming fi < Rand R, > Rk 


Gk Ry — 


R 
vers: Ai =" = ae, SO vers 4A (164) 


9. Assuming A, < 4A and diy << Se 


ra 1 ae 
R vers tA — Ry vers Ai (165) 


Ri a 
vers $A — Vers Ai 


3. Assuming Ai < $A and AA 


AA’ 
GK = Ta; = cotta ie 
Gk 
Ba = Rat vere FA ett 
GK 
By = Be ers Bi a 


4, Assuming R, > R and AA’ 
GK = (R. — R) verszA 


cot.4Ai.= cobtA +. pi (169) 

The third assumption will usually secure most readily the 

desired curve. AA’ should be assumed as small as the nature 

of the case will allow, and A, should not be much smaller 
than $A. 

It is evidently not necessary that the new curve should be 
symmetrical; for having laid out the curve A' PH, the simple 
curve HB may then be used, or, if desirable, some compound 
curve HP'B' determined by an assumed value of BB’ not 


equal to AA’. 


134 FIELD ENGINEERING. 


These formule (154) to (169) are readily adapted to the 
case of substituting a compound for a simple curve when it 
is necessary to keep one tangent point fixed, but to move the 
other a certain distance in either direction on the tangent. 
For if in Figs. 55, 56, we draw a tangent at H, and make H 
the fixed point of tangent, it is evident that the central angle 
of the curve will then be AOH. The only change necessary, 
therefore, to adopt the formule to this case is to write A in 
place of +A, and to observe, instead of eqs. (154) (161), that 

Ai 4 Aze= A. 
Hrample.— Fig. 55. 
Let it = "TOM. 08 = Be" 


Assume AA’ = 260. Ayes 08" 2S. Age S 
Kq. (166) AA’ = 260. log 2.414973 
: cotsA, 2.90421 19° 
cotta 2.60509 21° 
.29912 log 9.475846 
rs GK “ 2.939127 
Eq. (167) 4A 42° ** vers 9.409688 
3384.07 3.529439 
R 1910.08 
ee R, 5294.15 D = say 1° 05’ 
Eq. (168) GA log 2.939127 
Ai 38° *“ vers 9.326314 
4100.27 3.612818 
R, 1193.88 D= 4° 48' 
AP 791.67 PH = 369.23 


177. Given, two curves joined by a common tangent 
to replace the tangent by a curve compounded with 


_ the given curves. Fig. 57. 


Let R, = BO; the radius of one curve, 
fits = AO; the radius of the other curve, > R,, 

1 = BA the common tangent, 
““ R; = PO, = P'O, the radius of connecting curve. 
““ As = POP’ the central angle of “ 4 
‘a= AO;P' and 6 = BOP. 


COMPOUND CURVES. 135 


In the diagram join 0,0; and draw 0,G parallel to BA. 
Then in the right-angled triangle 0; GO, we have, 


.. GO,  Bs—k 
cot 4 = rita be j (170) 
0:03 = pis preak tC al eS. (171) 
COS 7 §1nN 2 


which gives the distance between the centres of the given 
curves. 


ace sena Dewees 


Fra. 57. Wh 

We shall now assume the following geometrical truths, fll} 
which may be easily demonstrated. it 

If two circles intersect in one point, they intersect in two 
points; and the line joining the two points is the common 
chord. 

The common chord is perpendicular to the line joining the 
centres, and when produced it bisects the common tangents. 

If a third circle is drawn touching the two circles, a tangent 
to the third cirele, parallel to the common tangent, will have 
its tangent point on the common chord produced. 

Conversely, therefore, if the tangent BA be bisected at K, HI | 
and a line, KJ, drawn perpendicular to 0,03, KI will coincide | 
with the common chord produced, and the angle Anes 
AO,0, =?. If on KI we assume a point T through which 1 
it is desirable that the connecting curve should pass, then J is | 
the tangent point of a tangent parallel to BA; consequently i 
a line through J perpendicular to BA contains the required | 


eentre Ox. 


136 FIELD ENGINEERING. 


All I. Let p = HI = the perpendicular distance between the 
tangents. 

If in the diagram we join [A and JB, and produce the 
chords to intersect the given curves in P and P', then P and 
P’ are the points of compound curvature; and the lines PO, 
and P'Os produced will intersect JO, in the same point 0,; 
and the angles P'O,J = « and PO.I = f. 

In the triangle AJB the line KT bisects the base AB, and 
we have by Geom. Tab. I. 25. 


AI? + BI? = 24K? + 2KT? 
By eq. (56) AI = 2(R. — Rs) sin 4a 
i) BI = AR, = R,) sin 38 


AK =H ands /KI 2. 
S1n 2 


Seley 


. 4a — Rey sin? ta + 4(Ra — Ray sin? 48 = 4P + 


Dividing by 2 and putting vers « = 2 sin? ta and vers paa 
2 sin? 46 (Tab. IT. 46) 


9 
(fh, — Rs)? vers « + (Rz — Ri)? vers 6 = 4P +. os 
But by eq (57) 


{ 
Hi 
i (R. — Rs) vers a = (R, — R,) vers B = p (172) 


 p QRy = (Ry + R))= ye 4 


Hii} sin? 7 


P Dp 
| | From (172) 
Pp Pp 
= ——_ ; So sve a 174 
vers @ a vers fj jek (174) 


and from the figure 


These formule solve the problem when p is assumed, If 
desirable we may find a and ( independently of R., for in 


COMPOUND CURVES. 


the triangle AJB, JAB = 4a and IBA = 14; and since 
HK = p cot 2, 


AH 41 — HK l ; ty 
cot ja = aT = 2p — cot? (176) 


We AK Y ; 
fe = -— a == op + cot2 (177) 


II 


cot 46 


Il. In case a or f ts assumed, we have from the last equa- 
tion 
l l 


= —<~ = 178 
RES 2(cot +a + cot 7) cot 44 — cot 2) fa79) 


Ill. In case the radius Ry is assumed, then in the triangle 
0,0.03 we know all three sides ; for 0:02 = (Ry — fh), 


0,03 = (R2 — Ri), and 0,03 = Ba ey 
COS @ 


By Trig. (Table II. 31.) 


2 ‘ae 9 s— € 
vers Ae = He oe a ; 0320s) 
1 2 7 B) 


in which s = + sum of the three sides. 
Substituting values, and reducing, observing that, 


tai) (itd) maces 
(om : ern =f SOP AIAN 


and that (R; — R,) tan 7 = J, we have 


[? 
vers Ac = 5(Ry — Ri) (Ra — Ba) (179) 


In the same triangle. 


: : 030. 
sin 0,0,0. = sin Ae 60. 


But from the figure 0;0,0. = 7 — #, and taking the value 
of 0,0, from eq. (171). 


FIELD ENGINEERING. 


sin (i — 6) = (A, — Bs) sin As sin (180) 


We then find a@ from eq. (175) and p from (172). 


The angles « and # may be found otherwise, for by Trig 
“Tab. Il. 27) we have in the triangle 0,0,0; 


sin +(0, Os Oz as 0; 0, Oz) = eal COS Ao 
1 3 


or 


‘ ., &€—f\ _ (Rs — R,) cost costAg 
sin (90 —(@+ ae) a Rate 


*. Cos («+ £ 3") = COS?. Cos 4A, (181) 


which is a convenient formula when ¢ and A; are not too 


a—fp 


small. Having obtained ao 
ft 


we have 


a— p ie a—p 
9 PA gre (182) 


a= gAot 


For a constant value of / the Jess the difference of R; — R, 
the greater will be the value of the angle 7. When R; = R,, 
cot ¢ = 0 and ¢ = 90° and the tangent point T will be ona per- 
pendicular to BA drawn through the middle point A; and 
a= f. On the contrary, as (R,; — ft,) increases, 7 becomes 
less, and the foot, H, of the perpendicular HZ moves toward 
B, the tangent point of the curve of smaller radius R;. The 
distance HK = p cot ¢. The connecting curve is farthest 
from the tangent BA at J. To find the ordinate from BA to 
the curve at any other point, subtract from p the tangent 
offset for the length of curve from J to the ordinate in ques- 
tion. §115, eq. (89) may be used on flat curves with tolera- 
ble accuracy, even when the distance equals several hundred 
feet. 

IV. It is evident that in this problem R, must be greater 
than either R, or R;. As the centre O, is taken nearer the 


a 


a 


COMPOUND CURVES. 139 


linc 0,03, Rz grows less, and is a minimum when OQ, falls on 
the line 0,03. In this case we have A. = 180°, and 


R, = (Rk; + R, +0,0;); a minimum. (183) 


This limit must be regarded in assuming the value of Ry. 
Since 


0,0, aps 0:03 — (Lt. rx R,) wy (Re a Rs) == (Rs <7 Rf) 


a constant value, independent of R:, we infer that the centre 
OQ, is always on a hyperbola of which O,; and Qs are the foci; 
(R, — R,) equals the diameter on the axis joining the foci; 
and / equals the diameter at right angles to it, for in the tri- 
angle 0,4 0s, 


i? =O, 0g he aie (184) 


Example.—Fig. 57. 


Given: R, = 1482.69 Rs; = 1910.08 and / = 400. 

Assume p= 11.4 to find Ry, and £. 

Eq. (170) Rs — A, 477.39 log 2.678873 
l 400. ‘© 2.602060 


39° 57’ 84" log cot 0.076818 


i 
Eq. (1°78) i 39° 57' 34" sin 9.807701 
5 39° 57°34" sin? 9.615402 
D 11.4 log 1.056905 
* 217.64 <* 1441508 
419 «4.602060 
p “1.056905 
* - 3508.77 “3.545155 
R+R, 8842.77 
2) 6879.18 
oh R; 3439.59 (say) 8487.87 
Eq. (174) p 11.4 “© 4.056905 
R,—R; 1527.79 «3.184064 
a 7° 00' log vers 7.872841 
11.4 log 1.056905 


p 
R,— Rk, 2005.18 «© 3.802153 
p (nearly) 6° 07’ log vers 7.754752 


13° 07" 


140 FIELD ENGINEERING. 


Krample.—Fig 57. 


Given: R, = 1482.69, Rs; = 1910.08, and 7 = 400. 
Assume fy = 3487.87, to find Aa, 6,a and p. 
2 


Eq. (179) log 0.301036 


R,—R, 2005.18 © 3 3091538 
R,— Rs 1527.79 © 3184064 
“ 6.787247 
12 5.204120 
. me 18° 07' 22" log vers 8.416873 
Hq. (170) Rs— PR, 477.89 log 2.678873 
eae A00: © 9 602060 
ve i 39° 57’ 34" log cot 0.076818 
Eq. (180) j 39° 57' 34" —_ log sin 9.807701 
es 13°.07' 29” «9 356099 
R,— Rs 1527.79 log 3.184064 
log sin 2.347864 
tae 200. log 2.602060 
i— 33° 50' 39" —_log sin 9.745804 
oe B 6° 06’ 55” 
Eq. (175) a 7° 00! 27" 
Eq. (172) Ry ~Rs log 3.184064 
at 7° 00' 27" log vers 7.873309 
p 11.41 1.057373 


178. Given: a three-centred compound curve to replace 
the middle are by an arc of different radius. 

I. When the radius of the middle are is the greatest. 
Fig. 57. 

First find the length and direction of the common tangent 
AB. Let A». = central angle of the middle arc, k, = its 


radius, and #, and R; the radii of the other arcs. From eq. 
(179). 


L = V2(R. — Ri) (R2 — Rs) vers Aa (185) 


Then find ¢ by eq. (1'70), a and 7 by eqs. (181) (182), and p by 
eq. (172). 

For the new arc we may now assume a new value for p, or 
for Rs, or for a. Indicating the new values by an accent, if 
we assume p’ we proceed as in the last problem, using eqs. 
(173), etc. If we assume R., we use eq. (179), etc. If we 
assume a’, we use eq. (178). 


COMPOUND CURVES. 141 


Il. When the radius of the middle arc is the least of the 
three. Fig. 58. 

In this case the middle arc is within the other two pro- 
duced; and for the same values of R,R; and 0,03, the locus 


Fia. 58. 


of the centre 0; is the opposite branch of the hyperbola found 
in §177. When the centre O» falls on the line 0,0:, As = 


180°, and 
R, = (Rs + Bi — 0102), a maximum. (186) 


Analogous to eq. (185), we have 
i= V2(R, — Re) (Rs oa Rs) vers Aa (187) 
which gives the length of the common tangent YZ. 


We then have the values of ¢ and of 0,03 by eqs. (170) (171), 
and of a and / by egs. (181) (182), and analogous to eq. (172), 


p = (R, — Rs) vers @ = (R;—R:) vers (188) 
in which p is the perpendicular distance HJ between parallel 


tangents. 
For the new are we may now assume a new value for p, for 


R,, or for e. Indicating the new values by an accent, if we 
assume p’, we have, analogous to eq. (178) 


FIELD ENGINEERING. 


ia eee VI is (a Baie ae 7) Gem 


and from eq. (188) 


vers = at ; vers-9' = oe (190) 
If we assume R,’, we have, analogous to eq. (179), 


5 Ue b? 
vers Ag = 2h — 1B, — Be) (191) 


and we find a and f by eqs. (181) (182), and p’ by eq. (188). 


Ill. When the radius of the middle are has an interaedi- 
ate value, compared with the other radii. Fig. 59. 


In this case, whatever be the value of hz, we have 
| 0:0, + 020; = (Rs Ser Re) + (fe Te fi) = (Rs he Rj} 


a constant value independent of R,; hence we infer tha) the 
locus of Oz is an ellipse, of which O; and Q; are the foci, and 
(R; — Rf) equal to the transverse axis. 

Let / = QQ = the conjugate axis, and let ¢ = Q0,0, = 
Q0;,03. 


Produce 0;Q to G, making QG = 0,Q, and join GO, 


4 


COMPOUND CURVES. 


Then by similar triangles GO, is perpendicular to 010s, and 
GO, = 1; and in the right-angled triangle GO30; 


F. agrinstr Oo l 
phi Od. wpm rd 192 
sant =o. — Ri Fh, (19%) 


0,0; —(R; — Ri) cost = ¢ cot t (193) 


Analogous to eqs. (189) and (187), we have 


1 = V &R; — Rs) (Ra — Bh) vers Az (194) 


which may also be derived from the triangles 0,0,0; and 


0:03Q. 
Let « = 0203;0;, and 6 = 020103 
Then 
] 2° oe vi — R ° ° . 
sin wa = “ a sin Az = aa tan 7. sin Az (195) 


From the figure 6 = 42— & (196) 


In the diagram produce the lime 0;0, and it will intersect 
‘e it cuts the inner 


all the arcs. At the points Z and Y, wher 
and outer arcs, draw tangent lines perpendicular to O30\. 
Draw the radius O.J parallel to 0:0, and the tangent line 


IL at I. 
Let g = ZY and p = ZL = HI 


Then by the theory of parallel tangents, $137, the point J is 
on the chord PZ produced, and it 1s also on the chord P’Y; 


and we have 
p = ZL =(Rz — R,) vers B. (197) 


q-p= LY = (Rf; — Ry) vers a (198) 


and g equals the swm of these. But g = ZY is the shortest 
distance between the inner and outer arcs, and has a constant 
value independent of &:. If we assume R, = (Rk; + #,) the 
centre QO. will be at Q, anda = fp = i,and p=4q. Making 
these substitutions above, 


q = (R; — R,) vers 2. (199) 


Also, from the figure, 


144 FIELD ENGINEERING. 


ZY = 0;3Y — 0:2 —.0;0s, 
i or, 


q peed R; ae R, — 0, Os (200) 
In the triangle ZTY we have by Geom. Tab. I. 26, 


ZI? = IY? + ZY* — 24Y (ZY — ZL) 


or 
ZY — 2Z2Y.ZL = IY? — Zl? 
Now, 
ZI? = 4(f, — Ry)? sin? 46 = 2R, — R,)* vers p 
ly? a 4(Rs — Re)? sin? 4a = 2(fs = Re) vers @ 
} Hence 


Hy ZI? = &B— Ry) p and IY? = &R,— Ba) (q—p) 


Substituting these values, and solving for p, we have 


_ Us — Rs — 49) _ (Rs — R. — 49) 


Epa Bie yer 0,0; Kote, 
Also 
By = (Ra ~ 4) — p20 (202) 
For any other value of R,, we have 
| Bi = (By — 4) — p' 128 
| i Hence 
| By — Ry = 228 (p — p) (203) 


Aisi which gives the change in R, for a given change in the value 
of p 


Observe that as p diminishes R, increases and vice versa. 
Having determined the value of R,', we find p' by substitut- 
ing &,' forR, in eq. (201); and from eqs. (197) (198) we have 


vers f' = sy (204) 


(205) 


s 


COMPOUND CURVES. 145 


and the change in the points of compound curvature is found 


by (6 — fi’) and (@ — @). 


Remark.—When Rz = +(Rs + Ry), Ae = 2, a minimum, 
and the long chord PP’ is perpendicular to 0,03. When fz 
is greater than this, « is greater than /, and vice versa. What- 
aver be the value of Re, the long chord PP' always cuts the 
line 0,0; produced in the same point S, at a distance from Zof 


ZS = R, vers 7; 


or from O, of 


OS — R, COS Of 


This item will be found useful in solving the problem 


graphically. 


Hzample. 

781.84 
1375.40 
1910.08 

11.30 


Rs — Rs 534.68 
Ry — fi 593.56 


me 
Tl Ut 


Ae 
rae l 458.27 
(192) Rs — hi 1128.24 
2 
(193) 1 
R; — Rk, 


0,0; 1080.98 


Ae 
se a 
(196) fp 
(203) 0,0; 
(200) q -97.26 
0:03 
q 


9—p 11.30 
Ra’ — fs 119.78 


1495.18 (say) 1494.95 for 3° 50’ curve. 


agi: 
=3j47)10'A5) Als. = 482 
Ds =. 3" 00' 
log 0.301030 
«© 2.728094 
‘¢ 2.773465 
48° log vers 9.519657 
2) 5.322246 
log 2.661123 


** 3.052402 
23° 57’ 55” log sin 9.608721 


23° 57’ 55” log cos 9.960847 
log 3.052402 


log ¥ 3.013249 


log 2.773465 
48° log sin 9.871073 


log * 2.644538 
25° 19! 52" log sin 9.631289 
22° 40’ 08" 


log 3.013249 
1.987934 


1.025315 
log 1.053078 
** 2.078393 


146 FIELD ENGINEERING. 


(201) R; — Rx’ —4q 366.50 log 2.56407 
Boe “© 1.025815 
q et ee 
23 p 34.57 “1.588759 
(197) Reis ge T18.41 «2.853157 
gp 17° 55’ ~— log vers 8 685602 
(198) roe 62.69 log 1.797198 
Rees ee alos ke 2.618184 
a’ 31° 54’ ~—log vers 9.179014 
na 49° 49’ 
Hi erat PP = 21689 
Boe Bs dean eo PP sp eebaald 


The practical difficulty in changing the middle arc of three 
centred curves lies in the difference of measurement that 
ensues. Thus, in the last problem, although the total central 
angle is the same, the new curve is 6.56 feet shorter than the 
original, making a fractional station at P'’. If the change is 
made during the location, it is well tu re-run the last arc from 
P'” to the tangent following, so as to eliminate the fractional 
station from the curve. 


TURNOUTS. 147 


CHAPTER VII. 
TURNOUTS. 


179. A turnout is a curved track by which a car may 
leave the main track for another. At the point where the 
outer rail of the turnout crosses the rail of the main track a 
frog is introduced which allows the flanges of the wheels to 
pass the rails. A frog consists essentially of a solid block of 
iron or steel having two straight channels crossing each other 
on the upper surface, in which the flanges of 
the wheels pass. The triangular portion of the 
upper surface formed by the channels is called 
the tongue of the frog, and the angle which the 
channels make with each other is called the frog- 
ungle. Every railroad is provided with a set of \ 
frogs of different angles, from which may be i 
selected one best adapted to any particular case. Fia, 60, 

The frogs may be designated by their angles, 
but it is customary to designate them by numbers expressing 
the ratio of the bisecting line FC of the tongue to the base 
line ad, Fig. 60. Observe that ¥ is at the intersection of the 
edges produced, ‘and not at the blunt. point of the tongue. 

In the triangle afC, 
sue. = cot 4aFb 

ad : 
and if we let 2 = the number of the frog, and # = the frog 
angle, then 


<b 
¢ 

| 

| 

| 


On some roads, however, the frogs are numbered arbitrarily, 
or according to their length in feet, while on others they are 
designated by letters of the alphabet. In any case the true 
number (x) of a frog may be determined by the above for- 


mula. 


148 FIELD ENGINEERING. 


The first rail of the turnout is common to both tracks, and 
is called the switch-rail. It has one end free, so as to be shift- 
ed from one track to the other as required; the free end, D 
(Fig. 61), is called the point of switch. The tangent point of 
the turnout, at A, is called the heel of switch, and the distance, 
AD, is the length of switch. The switch-rail should be several 
feet longer than AD, and the excess be spiked down in the 
line of the main track back of the point A. Then if the point 
D is thrown over to meet the rail of the turnout at A, the switch 
rail is sprung into an arc, which coincides with the arc of the 
turnout, provided that the ,ength of switch AD has been prop- 
erly taken. The distance DA through which the point moves 
is called the throw of the switch. It varies on different roads 
from 44 to 6 inches, but is usually made about 5 inches, or 0.42 
feet. A turnout should be a simple curve from the heel of the 
switch to the point of the frog. 


180. Given: a main track, straight, and a frog angle F, to 
determine the distance BF, on the main track from the heel of 
switch to point of frog, the radius, ¥, of the centre line of the turn- 
out, the length of chord af, and the proper length of switch AD. 
Fig. 61. 


Fie. 61. 


Let C be the centre of the turnout. 
““  F= the frog angle, HFT = FCB. 
‘* —g = the gauge of track AB. 
Sry E Oty Se PAGIUS, AG = $C, 
“«“ DK = the throw of switch. 


Then the radius of the gauge side of the outer rail is (7 + 49), 
and we have 


TURNOUTS. 149 


AB = FC. vers FCB 


or, 
g = (r+ 4g) vers F 
whence 
: oe ee: 
(7 + 39) = SG (207) 
The angle AFB=%3Ff 
and BF = ABcot AFB = g.cot4f (208) 


Again, in the triangle YCB 
BF = FC: sin FCB = (r+ }y) sin F - (209) 
The chord af is evidently 
af = 2rsintF (210) 
Similar to eq. (207), we have 
vers ACD = es = ay 


But since the inside rail has the same throw, while its radius 
is (r — 3g), we may, if convenient, drop the 4g, and hence the 
length of switch is 


AD =7r.sin ACD (211) 


The degree of curve corresponding to 7 is found from Table 
IV., or by eq. (17), and the centre line of the turnout may be 
located by transit deflections from the tangent point a, using 
chords of 20 or 25 feet -+ the correction found in S$ 106, 107; 
or the deflection for a 20-foot chord may be calculated at once 
by 


sin (dso) = —— (212) 


181. Simple as these formule are, they may be rendered 
still more convenient by introducing the number of the 
frog, 7. By eq. (206) we have cot 4F = 2n, which substi- 
tuted in eq. (208) gives 

BF = 2gn (218) 


Drawing the chord AF’ to the outer rail, 


AF = VAB?+ BF? =gV1+4n (214) 


150 FIELD ENGINEERING. 


Make BA’ = ABand join #4’, then by similar triangles, 
AA'F' and AFC, 


CAAA AP SS ale at 


whence 
AT? 
fhe = Stat 
or (7 + 39) = 49 (1 + 4n’) (215) 
whence Pe 2on = DR Sn (216) 


The chord af to the arc of the centre line is to AF’ as 7 is to 


AR? 
iO 4 : h _ ES ae 
(r + 39); hence af 2 iy 
eqs. (214) (215) we have 


, and substituting values from 


V1 + 4n? 
Assuming that, for small angles, the tangent offsets vary as 


the squares of their distances from the tangent point, which 
will lead to no material error in this case; 


(217) 


AB: DK :: BF?: AD? 


whence AD=BFy/ ee 
AB (218) 
or AD = y4n? 9. DK = V2. DK 


It is not necessary to determine the degree of curve in order 
to locate the turnout, for having fixed the position of BF, the 
position of af is found by laying off Ba, and Ff, each equal to 
2g. Whatever be the length of the chord af, found by eq. 
(217) or (210), its middle ordinate is always ig, and the ordin- 
nates at the quarter points, $. 4g = 3g. Thus for the stan- 
dard gauge of 4.708 the middle ordinate is 1.177, and the side 
ordinates 0.883. 

By the preceding formulw Table XI. has been calculated, 
which gives the required parts of a turnout for various frogs 
when the gauge is 4 feet 8! inches and the throw 5 inches; 
uso for a gauge of 3 feet and throw of 4 inches. For any 
other throw, only AD must be calculated. For a different 
gauge the engineer will do well to construct a similar table. 
adapted to the frogs used on the road. 


4 


TURNOUTS. La) 


In the table the frog angle 1s given to seconds, in order that 
the results may agree, whether found by equations in §180 
$181; but in practice the nearest minute is sufficiently 
exact. The frogs most used for single turnouts are those 
from No. 7 to No. 9, inclusive. 


or 


182. In case of a double turnout from the same switch, 


1..ree frogs are required, as at PV F' and I", Fig. 62., and the 
Ss J ’ 


Fig. 62. 


switch is called a three-throw switch, because its point takes 
three positions. The frogs #’ and F" are usually alike, and 
placed exactly opposite each other in the main track. The 
other frog Fis placed on the centre line of the main track. 
Its angle 7” and its distance from 4 are now to be determined 


in terms of /. 


In the figure we have vers Ra = ae or 
vers 32" = 5; a (219) 
27+ 439) 
The distance ak" = (r+ 4g) sin 3h" (220) 
aF" =r. tan3Fh" (221) 


also 
All the parts of the turnout required to locate the frogs /’ 
and FF are calculated by the formule in the preceding sec- 
tions, or are taken from Table XI. | 
If we let »’” = the number of the frog F''’, then by eq.(206) | 

{ 


tan 47" = os which substituted in eq. (221) gives i 


CH Se 
| 


& 


sy eee ee (222) | 


ee Ont. 


%~ 


152 FIELD ENGINEERING, 


Also, in the triangle aF'"C, 


al" = Vr-+ yg? — 7 = Vo(r +49) (223) 


Equating these and replacing r by 2gn?, we obtain 


Nims Mee (224) 
Qn? +3 
If we neglect the 4, we have 
(approx.) n' =" —  wWrin (225) 
V3 


tii Kxample.—If F = F' =6° 44, or n=7n' = 8.5, then ” = 
HH 6.0 + or fF” = 9° 32’. 


8 at hand of the angle or number given 
by eq. (219) or (225), we may select one as nearly like it as pos- 
sible, and locate the turnout as a compound curv 
vided that #”” is less than 2F' Fig. 63. 


} 183. In case no frog i: 


ey pro- 


Fig. 63. 


Let r" = O"a, andr = r' = Of = Cf" 


Then analogous to the equations of § 180, 


" Gy 4g 
BAG od 49 
* tla 4 exec eam (227) 


ah” = (r' + 39) sin dF" = r” tan $f" (228) 


| 
TURNOUTS. 


The length of the switch, by eq. (218), is 
AD = V2r" DK 


The curvature of the rail between the frogs 7” and F is 
F’'CK = (F — $F). 

Draw the chord #"’ fF’ and the perpendicular #’’L; then-the 
angle LFF" = F— y(h—3P") =F + 4f"); and since 
LF" = 49; 


; “up +9 
ite = sin H+ 4F") (229) 
LF =%g.cot4(PF+4F") (230) 
wy Le 6 
/ at a 
(r + 49) a sin + (F +8 4P") (231) 
Example.—Let F = 6° 44 F" = 10° 24 

Eq. (226) ig 2.354 log 0.371806 
4H" 5° 12' log exs 7.616224 
si r 569.616 2.155582 
Kq. (228) 4 Pf" 5° 12’ log tan 8.959075 
aF" 51.839 1.714657 


log 0.371806 
log sin 9.016824 


Kg. (229) 29 2.354 Pras: 
MF + 4F") B° 58 


Ss FF 22.645 
Eq. 81) 17 — 4") 


Ar + 49) 1692.432 
r 843.862 


0° 46’ log sin 8.126471 


3.228511 


When n” > . 707, 7 will be less than 7’. Should F’ not 
equal F, (F’" being given), then 7’ and ZL’ F” must be calculated 
also, by substituting 7’ for Win eqs. (230) and (281). 


184. From the same switch in a straight track tt 1s required 
to lay two turnouts on the same side. Fig. 64. 

If we assume #”’ = F, and that these two frogs shall be 
opposite each other, we calculate all the distances of the first 
turnout for the angle /# (or number 7) by § 180, 181, whence 
we have the radius 7 = Ca. 


1.354982. 


154 FIELD ENGINEERING. 


: | Let r' = C'a, the radius of the centre line of the second 
, turnout. The angle ACF’ = F, and since #" = F, the angle 
CF'C'= F, and the triangle CF"C’ is isosceles, and C'F’ = 
O'C.. But C'F" = C’A = 340A. 


or (7 + 49) = 4(r + 49) - (282) 
ee r = 47 — Wy) (238) 


C' 
Fic. 64. 


To calculate the remaining frog at #7", we have from eq. 
(207) 


owed | eto foe 
vers #"" — ge (234) 
4 or from eq. (216) 
Wp) et 
| nia (285) 
i | 2g 
| BF" = (r' + 49) sin F = 2gn" (236) 
1) ‘ ek ar" 
ii Bp OP Osin eh he cee (237) 
i V1 + 4n'? 
| and since AO'F" = 2F, ; 
af = oF sin dt (238) 


The length of switch may be calculated by either 7 or 2", 
since for 7’, which is about 47, the throw of switch is double, 

thus giving practically identical results. 
| If we compare the values of F’” as obtained by eqs. (234) 
and (219), we shall find them almost identical for given values 


. 


| 


TURNOUTS. 


of F'and g; and since this may also be proved analytically by 
assuming that vers +#’" = ¢ vers F'", which is very nearly 
true for ordinary values of #'", we conclude that a set of frogs 
(F = F', and #") which is adapted to a double turnout in i) 
opposite directions from a straight line (as in Fig. 62) is also | 
adapted to a double turnout on one side (as in Fig. 64), the 
curves being simple curves in every Case. But this being 
true, the set is also adapted to a double turnout in opposite 
directions from any curved track the radius of which is not 
less than 7 as given for F, since any such case is intermediate 
between the two cases named. When, ‘therefore, a certain 
frog, F, is adopted for general use on any road, another frog 
should also be adopted, whose angle, F", is determined by 
eq. (219), or whose number is determined by eq. (229). 
Thus, if 7 = 6° 44, or n = 84, then F" should be 9° 32’, or 
Re = 0. 


185. In case no frog is at hand of the angle or number given 
by eqs. (234) (2385), we may select one as near the same angle 
as possible, and, calling this #’", calculate the distance BF" 
and the radius CO" F" (Fig. 65) as for a single turnout; $ 180. 


Fia, 65. 


Then assuming any other frog #’’, whether equal to F or not, 
it is required to find the chord FF’, and the radius 0'F" of 
the arc F’F’. The point #’ may fall either side of the radius 
OF. according to the values given to #” and #". 

a. In case F' falls beyond the radius CF, we will assume 
first, that the entire rail from B to F’ is laid with the same 
radius BO, and centre 0. (This investigation also applies to 
the case when F’ falls between B and the line CF’) 

In the diagzam (Fig. 65) draw CF". We then have 


FIELD ENGINEERING. 


BF" BR" 
UAL Besa (ee. 9 
tan BOP” = Bo = (289) 


and 
GF" = (r — 49) exsec BCF"’ (240) 


In the triangle #'"CF’. 
F°C— F'C : F'C+F'C:: tant" F'C— F'F'C) 
> cot #" CF" 
Now, since C'’F'C = F', and BC" F"' = F", 
ot de ee ee 


and 
Pod Om FR GEO = FORO ms (Fh — BCF") 
Letting =" C OH Cre Cee BOR) 


and subtracting, we have 
F'RC-—FF'C=F'+U0 
Hence the above proportion may be written 


GF" :2BC-+ GF” :: tan 4" + U): cot $F" CF’ 
whence 


po wes tan (F’ +0) (241) 


(Since BOF" + FCF" = BCF"', and we know the radius 
BC, the chord or arc BF’ is easily obtained, which fixes the 
position of the frog #’'; and the problem may end here, 
frequently, in practice.) 

Now in the same triangle CF", the half sum of F'' F'C 
and F''F'''C is 90° — 4F'"CF"; while, as we have just seen, 
the half difference is }(F'' +- U); and by subtracting we have 
the less, or 


F'r'C=90 — UF’ LU+ POF) (242) 


F'Osin FOF" 
sin F'P''0 


cot 3#"CF’ = 


Now 4 hie tae 


BC. sin FCF" 


oe Ee son UHL (Lt RY OF 


(243) 


“ae 


TURNOUTS. 


To find the angle F'''C' F''; produce the line #''C’' in the dia- 
gram to intersect the line BO at A. Then the two triangles 
HO'C' and KOF' have the angle A common, and the sum of 
the other angles will be equal; that is, 


KOO -- O O'R = KO --— Cr kK 


or FOL ORs BOR hek i 
and since BCF' = BOF" + FCF’ i 
BO Gs POR + Fe — 0 (244) Wi 


If we denote the radius /''C’ by » + 49 
, 4h" F" 
/ 1 — 
m= SP Or” 
Example.—Given: the three frogs 7 = 6° 43' 59", #” = | 
6° 01’ 32", and #" = 8° 47 51" to lay a double turnout on one i, 
side of a straight track. Fig. 60. il 


(245) 


By Tab. XI. BF = 80.0386 r = 680.806 AD = 23.82 
BF" = 61.204 7" = 397.826 


Eq. (239) BF" 61.204 log 1.786779 Wall 
(7 — 39) 677.952 « 2°831199 ui 
BOF" 5° 09' 88" log tan 8.955580 ih 
Eq. (240) BOF" 5° 09' 38" log exsec 7.609587 | 
(ry — 49) 677.952 log 2.831199 | 
GR’ * 2:760 “0.440786 ul 
Eq. (241) 2BC-+ GF") 1358.664 «3.133112 H 
(U = 8° 38’ 13") 2.692326 NW 
34(#” + U) 4° 49' 52".5 log tan 8.926968 He 
te Jeane ees Ha 
(F" OF’) 1°22'35" “© cot 1.619294 I} 
Eq. (243) F" OF' 9° 45'10" «sin 8.681481 | 
r—4g 677.952 2.831199 He | 
Se a oe Se Bia 4 
1.512680 ) 
MF’ + U+ F'OF’) 6° 12'27".5 ** cos 9.997446 iH 
F’F’ 82,752 1.515234 i 


Eq. (245) 4F"C'F' ; 9° 34! 14.5 6“ sin 8 651781 | | | 
vin Ar’ +49) 780.219 2.863453 i] 
r 


158 FIELD ENGINEERING. 


C 


b. We assume, secondiy, that the middle track is straight 
beyond F, and tangent to the curve at F. Fig. 66. 

Then whenever the value of /”" 1s dess than that given by 
eq. (234), the arc AF”, produced with the same radius AC", 
will intersect the straight rail H#” at some point /#”, and the 
frog angles f'and fF" will be equal. 


Fia. 66. 


For the straight rail H#” produced backwards, passes 
through the point A, making an angle #’ with the main 
track, since the triangles CBF and CHA are equal, and AH 
= BF. Now any circle, tangent to the main rail at <A, 
will intersect the line AH in some point /’, and since AF’ is 
the chord of the are, the angle at #7’ equals the angle at A, 
which is #. Hence # = #"; and the angle AC" F” = 2F. 

The length of the chord AF" is 


AF’ = 2AC" sin F (246) 
| | | | The chord #"F" = 2F'"C" sin}(F"C"F’) 
I, = 2AC" sin }(2F — FP") 
ti Hence, F'F" = Ar" + 4g) sin(F¥—4F") (247) 


\ | Hrample.—Let fF’ = F = 6° 4359" and F'” = 8° 47' 51” 
By Table XI. r” = 397.826 _ 


Eq. (247) 27" + 49) = 800.360 log 2.903285 
Fo—4F" 2° 20'03".5 log sin 8.609915 
F"F" 32.60 1.513200 


If the frog ¥’ 1s required to be different from F, then the 
inside curve must be compounded at F', giving other valués 
to the length and radius of the are #'" F’. 


ES 


TURNOUTS. 159 


c. We assume, thirdly, that the curve of the mzddle track is 
reversed at F. Fig. 67. 

In the diagram, let Q be the centre of the reversed portion, 
and F" the proper position of the frog F’, and CC’ the centre 
of the required arc FF". Then Q ison the radial line CF, 
produced, and 0’ is on the radial line #0" produced. Join il 
FQ and F'Q, and produce (F'" to intersect these lines in L | 
and M respectively. Also join FQ, and denote the angle i 
LF'’Q by U and the angle F'QF” by @. at 


Fig. 67. 


In the triangle F'"Q we know #"F = BF — BF", and the 
side FQ is given; and the included angle F' FQ =90° + F. Hl 
Hence we may calculate (Tab. II. 25) the angle F' QF and the | 
side FQ. 

The triangle CCL gives the angle at L = #” — F’, and the 
triangle LQ gives LF" G@ = L — Por 

U= F’ —F— F'QF (248) 


In the triangle /' QF" we have | 
F'Q—F'Q: FQL FQ: tany(F'F"@ — F’'F'Q) i} 

: cot 4(F' QF") | 

But 2’ FQ = F'F"L4+ Uand F’F'Q=Ff 'F'N — F’, and | 
since FF" N = I'F'L, we have by subtraction, 
F'P'Q — F’F'Q=U+F' HH 


pO reed aa: 
Hence cot 49 = F'Q-F'Q tan 4(U + F’) (249) 


160 FIELD ENGINEERING. 


(Now the angle FQF’ = Q — F’'' QF, and is subtended by the 
chord HF", which is therefore easily found, and serves to 
locate the frog #’’, and frequently this is all that will be 
required. ) 

In the triangle #”'’'Q/’, thehalf sum of QF’ F’ and QOH Ff" 
is 90° — 4Q, while, as we have just seen, the half difference is 
34,\U-+ F’); hence by adding, we have the greater, or 


QF’ F' = 90 4+ KU + FQ 


Pi ae, eee sin Q re 
ee seed |e © cos (UF — (250) 
The triangle C’F'M gives F''C'F"’ = F' — M, while the 
triangle #""MQ gives M= U + Q; hence FP" C'F' = F' — 
(U + Q); and denoting the radius (''F" by r’ + 4g, 


4" F' 


cb SS cre (Ff — U—@Q) Gey 


Example.—Let F = F" = 6° 43' 59", F’ = 8° 4Y7’ 51", and 
FQ = 953.012. Then by Tab. XL, BF = 80.036 and BF’ = 
61.204; hence #’’"# = 18.832; and the included angle is 
96° 43' 59". 

Solving the triangle 77'"Q we find LOE Or Ae 
FF"Q = 82° 08' 48", and #"Q = 955.402. Now it © a 
FQ + 9 = 957.720. 


(249) F’'Q+ FQ 1913.122 log 3.281748 
TL) See ye 


Q 2,318 © 0 365113 

(U 0° 56’ 34") “2 916630 

KU + F’) 3° 50’ 16".5 log tan 8.826231 

1Q 1° 02° 08".4 “ cot 1.742861 

(250) Q 2° 04’ 16".8 <* sin 8.558033 

HU+ F'— Q) ~ 2° 48’ 08".1  “ cos 9.999480 

8.558553 

F’'Q 957.720 log 2.981289 

ES F'F' — 34.633 1.539892 

(251) 4’ —-U— @Q) 1° 51’ 34".1 log sin 8.511191 

Q(r' + 49) 1068.32 3.028701 
581.81 


TUBNOUTS. 161 


186. Given: a main track, curved, and a frog- -angle F, to 
locate a turnout on the inside of the curve. Fig. 68. 
Let R= Oa = radius of main track. 
« » = Ca = radius of turnout. 
« H— OFO = the frog angle. 


In the diagram draw the chord AF’ and produce it to inter- 
sect the outer rail at G; and draw FO and GO. Since the 
chords AF and AG coincide, and the radii AC and AO 


Fic. 68. 


coincide, the chords subtend equal angles at CO and O respec- 
tively, and GO is parallel to FC. (See $137.) Hence, FOG 
— (CFO =F. Let 6 = the angle FOA. 

In the triangle FOA, 9 = GFO — FAO = GFO — FGO,; 
and in the triangle @FO, GO4+ FO: GO— FO :: tan HGFO 


+ FGO) : tan (GFO — FGO), or 2R : g:: cot iF: tan 49 
: g gn 
2 fan3o-= oR cot 37 = PB (252) 
In the triangle CFO, 
r+ io) = (R—-3 = ein 8 O59) 
+i) =k - WD FPS (253) 


In the triangle BOF, 
BF = (R — iy) sin 39 (254) 


In the triangle aCf, 
af = 2r sin WF+ f) 


162 FIELD ENGINEERING. 


The length of switch AD, for a given throw DK, may be 
found thus: from Table IV. take the tangent offsets, ¢ and f, 
corresponding to # and 7 respectively, and assuming that the 
offsets may vary as the squares of their distances from the 
tangent point, we have 


Bes aad see & Ws Gods OR Ui Ge BE. 
t—¢ 
This result is practically the same as that found for length 
of switch in a turnout from a straight line with the same frog, 
when J is large. 
Heample.—Let R = 1482.69 and # = 6° 43’ 59". 


Eq. (252) ig 2.854 log 0.871806 
iP 


} 3° 21°59".5 log cot 1.230440 
log 1.602246 
R (Tab. IV.) ° 3156151 
19 1° 85'59".8 log tan 8.446095 
Eq. (254) 8 3°11'59"6  — **_ sin 8.746786 
P46 9° 55’ 58".6 6 9 936778 
9.510008 
R— ig 1480.3386 3.155438 
r-t4g 462.856 2. 665446 

r 460.502 
254) 9 log 0.801030 
(R—4g) 1430.336 “© 3.155438 
ase 1° 35'59".8 log sin 8.445924 
: BF 79.872 log 1.902392 
(255) Qr 921.004 2964269 
i(F + 6) 4° 57'59".8 log sin 8.937381 
af 79.734 log 1.901643 


The values of BF and af are found to be so nearly identical 
in this case with those determined in case of a turnout from a 
straight line, that the values given in Table XI may be used 
at once for ordinary values of R; and the degree of curve of the 
turnout in this problem is approaimately the sum of the degree 
of curve of the main track and the degree of curve given in 
Table XI. opposite #. Thus, in the example 4° + 8° 26’ = 
127526... 7. 7 = 461.7 idarls 


TURNOUTS. 163 


187. Given: a main track, curved, and a frog-angle F, to 
locate a turnout on the outside of the curve. Fig. 69. 

In the diagram draw the chord AF, and produce it to meet 
the inner rail at G; and draw /O and GO. The triangles 
OAF and OAG are both isosceles, and have the angles at A 
equal; hence they are similar, and FCA = AOG. Hence 
FOG = HFO=F. Let R= Oa, r = Ca, and 6 = FOA, 


Fia. 69. 


In the triangle FOA, 6 = OAG — AFO = FGO — GO; 
and in the triangle FOG; FO + GO: FO — GO:: tan 
4(FGO + GO) : tan 4(FGO — GFO), or 2h : g 3: cot 4H 


: tan 46 


: Lae OP Nee 25 
. tan 40 = oR cot 4+# = R (257) 
which is identical with (252). 
In the triangle CFO 
sin 6 
(7 + 49) = (Rk + 39) ain (= ® (258) 
In the triangle BOF, 
BF = AR -+ ig) sin 46 (259) 
In the triangle a@f, 
af = 27. sin 4(F'— 6) (260) 


For a given throw, the length of switch will be 


a an eae Se) 


fay 


164 FIELD ENGINEERING. 


| in which ¢ and ¢’ are the tangent offsets (Tab. IV.) corre- 
| sponding to # and 7, 

In this problem, as in the preceding, we may, for ordinary 
values of #2, assume the values for BF'and af given in Tab. XT. 
The degree of curve of this turnout is, approrimately, d — D, 
taking d from Tab. XI. and D from Tab. IV. corresponding 
to Rk. Should D = d, this turnout becomes a straight line; 


Fie. 70. 


and when D > d, or when # is less than 7 given in Tab. XI., 
the centre falls on the same side as O. Fig. 70. In this case, 
using the same notation, 46 is given by eq. (257). 


sin. 
(7. — 49) = (kh + ty) sind —F) (262) 
| Eq. (259) BF = &AR-+ 4g) sin 36 
| af = 2r sin 44 — F) (263) 


188. A tongue-switeclh is a short, stiff switch which; 
| when moved, revolves at the heel as on a pivot. When it is 
HI thrown over to the turnout track, it makes an abrupt angle 
with the main track, called the switch angle; but in this posi- 
ii tion it should be tangent to the turnout curve. .The use of 
this switch is generally confined to yards and warehouses, 
where but little space can be afforded, and where the motion 
of the cars is always slow. 


189. Given: a straight track, a frog-angle F, and the length 
and throw of a tongue-switch, to locate the turnout. 


TURNOUTS. 165 


Let AD be the length, and DK the throw of switch, and let 
S denote the switch-angle DAK. 


DK DIE 


Then sin S = “ap S° = 57.3 AD (264) 


(Compare § 86.) 
Let @ be the centre of the required turnout, and in the dia- 


gram draw OK and CF; also draw DG perpendicular to the 
straight track. Then DG = F: and in the triangle KGC, 
KOF = KGF —GKC, and since CKA isa right-angle, GKC 
eae OP = Lor a. : 

Draw the chord KF, and since the triangle KCF is isosceles, 
the angle C/K = 90° — 1(F — 8)... Now, CFI = 90° — F, 
hence by subtraction, KF = HF -+ 8). 


Fig. 71. 


If g denote the gauge, we know KI = g — DK; and in the 


right-angled triangle KIF, we have 
f= Ki. ct te+t 8): (265) 


KI As 
AP = nar er 8) (268) 


KF 
r + io Sra — 5) C8?) 


These equations are analogous to eqs. (229) (230) (231). 


190. Given: a double turnout with tongue- 
switch, from a straight track; to find the angle, F', of the 


middle frog. 


Assuming 7” = calculate (r -- 49) by the last equations. 
n the centre line of 


Since the rails of the turnouts intersect oO 


166 FIELD ENGINEERING. 


he straight track, as in Fig 63; if we substitute the value of 
Lf" F’, eq. (229) in eq. (281), we have 


OF 19) = 2 5in Ea) sin KP EP) 


and by Trig. Table II. 


With off89 7A) Be 
ia cos 4f'" — cos F 


Be ee 4g 
whence cos 4" = cos F + —~* 268 
If the angle of the middle frog to be used does not agree 


it) with #’” found by the last equation, the turnout will be com- 
Ht) pounded at 7’. 


191. Given: a straight track, the frog-angles I, F' and F", 
and the switch angle S, to locate a double turnout. 
Fig. 72. 


| 
| 
| 
| 
| 
1 Fia. 72, 


Assuming that #” shall be placed on the centre line of 
the straight track, let h be a point on the centre line at the 
point of switch. Then AK = 4g — DK; and since the angle 
Ff" is bisected by the centre line the necessary formule in this 
case are obtained from §189 by simply replacing by $F" 
and AJ by 44; and in the first. members [7 by AF" and r by 


rn’. This is obvious by the similarity of the figures, 


TURNOUT. 167% 


LK . cot 44" +8) (269) 


hk 
= 2 
— gin 444" +8) (270) 


II 


Hence AF" 


KF “" 


sighs eases Wy 


The location of the remaining frogs is a problem already 
discussed, § 188, ea. (229), etc. 


192. Given: a straight track, the frog angles F, F', F'", and 
the switch angle S, to locatea double turnout on one 
side. Fig. 73. 


Fic. 73. 


The frog Fis located by $189; but for the frog #'” we have 
evidently a double throw; hence eqs. (265) (266) (267) become 


IF" =(g — 2DK) cot (F'’ + 28) (272) 
KP =~ y(F -+ 28) oe) 
IT (274) 


K sido sin 3(#” — 28) 
To locate the remaining frog F': when F" falls beyond the 
line CF, there are three cases. 
a. The middle track reversed beyond F. 


We find the distance #'" #' by subtracting TF", eq. (272) from 
IF, eq. (265). after which the solution is identical with that 


given $185, @., Fig. 67. 


168 FIELD ENGINEERING. 


b. The middle track compounded at F. 


Let Q be the centre of the curve beyond F, and also let ss 
the angle #’Q/’"; and let U = the angle 0" F'"Q. 
Then by a course of reasoning analogous to that of case a, 
we derive . 
U=F"— F4+ F'QPr (275) 


cot 4Q = re te re tan 3#(U + F’) (276) 


Now since the radius 7’'Q is given, and the angle QF’ = 


Q@ — FQF", we readily determine the distance HF’ ‘, and so 
locate the frog F”’. 


In the triangle 7" QF", the half sum of QF" F’ and QE'F"’ 


is 90° — 4Q, while the half difference is 4(U -+ #”"); hence by 
subtraction we have the less, or 


F'F'Q = 90 —KU+ F'4+@ 


‘Tw ’ sin Q 
Hence LSS. eae HULEF LO (277) 


Join C'@, and the quadrilateral C’QF"' F" gives 
f"+Q= U0+ F'"C'F' 


hence #'"0'#”" = F’ — U-+ Q; and denoting the radius C'F" 
by r’ + 4g, we have 


mig = 


4H" fF" 
sin (F’— UF Q 


(278) 


Cor. Since the centre Q is assumed at pleasure, it may be 
made to coincide with the centre (0, and then the compound 
curve becomes a simple curve. Then also, the above formule 
will apply when 7” is such that the frog will come on the are 
JH. But as FQF" will be greater than Q, the difference 
FQF' will be negative, indicating that the distance HF’ is to 
be laid off backwards from H. 


c. The middle track straight beyond F, and tan- 
gent to the curve at F. Fig. 74. ; 

Let #” be the required position of the frog F'. A tangent 
to the curve at #’ makes an angle (#’ + F) with the main 
track, and a tangent at #” makes an angle of #’” with the 
same; hence the angle they make with each other is 


; 
> | 
TURNOUTS. 169 


(F'+ F— F"), and this is the curvature of the are 1" F", 
and equals the angle #'"C' F". 

Produce the straight line 7’ H backwards to G, and draw 
F’G perpendicular to it. Then #"G= FH — FF’. sin F, or 


F'G=g—F'F.sinFf —. 795 


Fig. 74. 


In the right-angled triangle #’GF'", the angle NG Ge 
F—4(F' 4 F— Ff") =F’ + FF’ — #). 


"a F'G 
and _ GF = F'F'.. cos 4(F’ + F" — FP) (281) 


Observe that GF" cannot be less than GH = F" F. cos F. 


193. Given: a turnout with a frog angle F, and the perpen. 
dicular distance p between the centre lines of the main and side 


tracks ; to find the radius x of the curve connecting the 
turnout with the side track. Fig. 75. 


170 TIELD ENGINEERING. 


Let the reversing point be taken at /, and let Q on CF’ pro- 
duced be the centre of the required curve, and draw QM per- 
pendicular to the main track. Then QM= QF =r — 4g; the 
point Wis the point of tangent, and the angle FQM = FL. 

Now JV being the intersection of the rail BF’ with the radius 
QM, we have MN= QF'vers F, but MN=p—g; hence 


The distance FN is evidently 
FN = (r — 49) sin & (283) 
and the chord to the centre line is 
fm = 2r sin $F (284) 


Should the distance F'N consume too much of the track, it may 
be lessened by introducing a short tangent at F, denoted by k; 
then by eq. (48) the radius will be shortened by an amount 
equal to &. cot 4F, and the distance /N will be shortened by &. 

Since the tangent & reduces the length of the tangent offset 
of the entire curve by k . sen F, we have for the new radius 7” 

p—g—ksn Pf 


ro Ee “vers # CoH) 


When 2’ is fixed by a limit, we obtain & by resolving eq. (285) 


SPRATT ayyvers a 


é sin 


(286) 


In case the main track is but sghtiy curved, we may at first 
assume it to be straight, and find 7 as above, eq. (282), and 
the degree of curve corresponding to 7; but this degree of 
curve must then be ¢ncreased or dimimshed by the degree of 
curve of the main track, according as the track is concave or 
convex toward Q. 


194. Given: the perpendicular distance p between the centre 
lines of a curved main track and a parallel side track, and the 
frog angle F of a turnout, to find the radius vr of the connecting 
curve, and the length FN, or fm, of the curve. Fig. 76, 


4) 


TURNOUTS. 


Let FN be the rail of the main track, and GM ihe rail of 
the siding, adjacent to each other; let O be the centre of the 
main track, and Q the centre of the connecting curve. Then 
the connecting curve will terminate at m, on the line OQ pro- 
duced, 

In the diagram draw MF, and produce it to intersect the 
rail MG@ at G, and join GO, FO, and FQ. , 

Let R =-radius of centre line of the main track; 7 = radius 
of centre line of the connecting curve; and @ = the angle wi 
FOM. © Wy 


Case a.—The siding outside the main track. Fig. ‘76. 


By similarity of the triangles GOM and FQM, GO is paral- 
lel to FQ, and the angle GOP = F; and by a process similar 
to that of $186, we have 


AY 
tan 109 = 55 iP cot 47 (287) I} 
hie 
A sin 6 , Hl 
r— 49 =k 39) in (FL 6) tees) | a 
FN = 2(R + jg) sin 49 (289) i} 
fm = 2”. sin 4+ 4) (290) : 


Case b.—The siding inside the main track, Fig. 77. 


By-a process entirely similar to $187, we have Ih 


zt Pp promi , Py 
tan 49 = aR» cot +# 


FIELD ENGINEERING. 


sin 0 


r—wWwa(h — 49) an (P= 8 (292) 
FPN = AR — 49) sin 46 (293) 
jm = 2r sin 47 — 9) - - (294) 


When 6 = F’in the last equations, sin (/ — 6) = 0, and r — 
}g is infinite, and the curve 7M becomes a straight line. 


Fie. 77. Fie. 78. 


When 6 > F, sin ( — 9) is negative, and the centre Q falls 


bf 


on the same side of the track as O, and we have 


sin @ 


r+ig=(hk- 49) Sin 0 — F) (295) 
fm = 27. sin 46 — Ff) (296) 
Equations (291) and (293) remain unchanged. 


195. To locate a crossing detween paraliel tracks. 
Fig. 78. 

When a turnout from one track enters a parallel track by 
means of another frog and switch, the whole is called a cross- 
ing. The frogs are alike, and the calculation for one end of 
the crossing answers for the other. §§180, 181. We have 
only to find the length of track between the two frogs. 

In the diagram let Af’ be one turnout, and A'/” the other, 
connected by the straight track #'G@. It is required to deter- 
mine the length #'G, or the distance NV measured on the 
main track from F to a perpendicular through 7’. Produc: 
ing the line #’@ to intersect the rail VF’ at H, we have two 


av 


TURNOUTS. 173 


right-angled triangles GFH and F' NH, having the common 
angle at H = F. Let p= the perpendicular distance between 
centre lines of main tracks, andg = gauge. Then Gi’= 9g, 
and fN.= (p.— g-) 


r@=Pru— Gut ~- GFretF 
sin # 
1 nike teeidoas i 
an P@= k= P-L g cot F (297) | 


a - Sk = A hs +a g 9 Al | 
So FN = NH — FH =(p—g) cot Ff on (298) vi 


When the main tracks are curved the distance F'G may be Hy 
calculated by the same formula (297) which gives a value only 
a fraction too small, but in laying the track the rail #’’@ must 
be curved to a radius which is to R of the main track as 


F'G: NF. 


196. When p is large, or the tracks are very wide apart, it 
will effect some saving of room to lay the crossing in the form 
of a reversed curve ; and the frogs being alike, the two 
arcs will be equal, and the point of reversed curve P will be Hi 
midway between Fand F"’. Fig. 79. yh 


In the diagram we have aPa' the centre line of the cross- 
ing, and PL the centre line between tracks; al = 4p, and 
aC —aC'=r. The radius 7 having been found by § 180 or 


& 181, we have iW 


vers aCP = ne (299) 


and PL=rsin.aCP 


174 FIELD ENGINEERING. 


1 The distance between frogs, /“V, measured on the main track 
is evidently 


FN = APL — BF) (301) 


in which BSF’ is determined by eqs. (209), (218), or by Tab. XI. 


197. To lay a crossing in the form of a reversed 
curve, when the parallel tracks are on a curve. Fig. 80. 


Let O be the centre of the main curve, Cand C’ the centres 
of the reversed curve. 
ut Then in the triangle COC' we know all three sides; for CO 
a =R+r;CC=r4+r,and0O= R+ p—r';; and the half 
. sum of the three sides iss = R+7-+ 4p. 
ill Denoting the angle COC’ by @, we have (Trig. Tab. II. 31) 
TH p(T +1 — 4p) 
Wil ves ® = (R48) (RED) 
| The angle g determines the length of the arc BW described 
Hil with the radius (& + 4g) and so fixes the position of the point 
| A’ from A. 

By a formula similar to the above, 


(302) 


10g — Pi — r+ 4p) 
vers 0'00 = (Rin eer) (303) 


TURNOUTS. 175 


The angle O’CO determines the length of the are aP 
described with the radius 7; the angle (p + C’'CO) = CCA’ 
determines the length of the arc Pa’, and P is the point of 
reversed curve. 

In this problem R is known, 7 is found by § 187, and 7 is 
found by $186, only observing that in this case the value of 
R must be increased by p. The frog angles /'and F’’ may be 
equal or otherwise, only taking care that the point P shall be 
included between the radii C'F" and CF. 

The angle FOC = 6 is given by eq. (257), and the angle 
F'0C' = 6' is given by eq. (252) (in which the value of £ is 
to be increased by p); hence the angle FOF’ = mp — (6+ 6), 
which determines the distance between the frogs, measured on 
the main track. 


198. To find the middle ordinate m, for 1 sta- 
tion, or 100 feet, on any curve, in terms of the degree of 
curve D. 

Referring to Fig. 4 we have in the right triangle AGH 


GH = GA. tan GAH 


But GA = +}AB = 40, and (Tab. L. 18) GAH =4AOB =34; 
hence 
M=3C. tan}ta (304) 


a general expression for the middle ordinate of any chord. 
If in this equation we make C = 100, A becomes D; and 
denoting the corresponding value of M by m, we have 


m = 4100 tan {1D (305) 
whence the rule, Multiply the nat. tangent of 4 the degree of 


curve by 100 and divide by 2. Thus the values of m in the 5th 
column of Tab. IV. have been calculated 


199. To find the middle ordinate for any chord in 
terms of the chord and radius 
Referring to Fig. 4 we have 


GH = OF — 0G = OE — VAO?— GA? 


ibe pal / Ge (<) (306) 


176 FIELD ENGINEERING. 


When CO = 100 we have for the middle ordinate of one 
station 
m= R— VR? — 2500 (307) 


For any subchord ¢, less than 100, we have for the middle 
ordinate, 


m, = R— j/ (2) | 


2 | pie Secrets 
m = kR— y/ (e+ =) (x—<) 


By adding se to the quantity under the radical in eq. (808) 


it becomes a perfect square, giving 
eC 


Mi = se nearly, (809) 


which is a very useful formula, although approximate. The 
error in m, does not exceed .002 for any subchord ¢ when the 
radius is greater than 800. On a 20° curve the error will be 
002 for a chord of 50 feet; and on a 40° curve the error in ™m 
will be only .003 fora chord of 33 feet. Hquation (809) is 
therefore practically correct in all cases for finding the middle 
ordinates of rails. Table XII. is calculated by eq. (808). 


200. Curving Rails. Before any rail is spiked to its 
place in a curve, it must be evenly bent from end to end, so 
that it will assume the proper curvature when lying free. 
The bending may be done by using sledges, but is best accom- 
plished, especially for turnouts and other sharp curves, by 
using a bending machine made especially for this purpose. 

The proper curvature of a rail is tested by measuring Its 
middle ordinate from a small cord stretched from end to 
end and touching the side of the rail-head, .The cord should 
also be stretched from the middle point of the rail to either 
end, and the middle ordinate of each half length measured, 
to test the wnzformity of curvature. 

From the last equation it appears that, with a given radius, 
the middie ordinate varies nearly as the square of the chord. 


TURNOUTS. 


We may therefore find the middle ordinate of a rail whose 
length is ¢ by the proportion 


(100)? : @ i: ms Mm 


or Mm, = cm yearly (310) 
eT } 
in which m is obtained from Tab. IV., col. 5, for the given 
radius or degree of curve. 

Example.—What is the middle ordinate of a 30 ft. rail 
when curved for a 20° curve? 

900 X 4.374 
Eq. (310 ee ee IN, 
When a long rail is bent for a sharp. curve, observe that ¢ is 
the length of the chord of the rail—not of the rail itself. 

For the chord of half a rail the middle ordinate is one-fourth 
the middle ordinate of the whole rail. Thus, in the above ex- 
ample it would be .099 or 14% inches. 

Instead of using the chord of the whole rail, it may be.more 
convenient to assume a chord shorter than the rail, especially 
when the chord is not an exact number of feet, knotting the 
string to the length assumed, and applying it to different por- 
tions of the rail successively. 


201. Elevation of the outer rail on curves. 

When a car passes around a curve, a centrifugal force is 
developed which presses the flanges of the wheels against the 
outer rail. This force acts horizontally, and varies as the 
square of the velocity, and inversely as the radius of the 
curve. Denoting the centrifugal force by f, we have from the 


wv? 


theory of mechanics f= 39,166 R’ in which 7 = weight of 
fe ay 


loaded car in pounds, v= velocity in feet per second, and 
R=radius of curve in feet. 

In Fig. 81, let ab represent a level line at right angles to the 
track, let @ and ¢ be the tops of rails on a curve, let de = ¢e= 
elevation of outer rail ¢, and let the point d be the centre of 
gravity of the car. The force f acts in the direction ad, and 
if f’ = the component of f in the direction ac, then 


fis fii ab: ae. 


178 FIELD ENGINEERING. 


The weight w, resting on the inclined plane ac, developes a 
component in the direction ca, and denoting this by #', we 
have by similar triangles, 


wm :w:: bc: ac. 


7 
Fie. 81. 


Since equilibrium requires that w’ shall equal dir ’, we have after 


dividing one proportion by the other a ithe , or f= —— 


Equating this value of f with that given above we find, 


ook CaS 
~ 32.166 R 
But a = V ae — e’, and ac = distance between rail centres = 
gauge + one rail head = g + 0.188. Also v= sae V, if V de- 


note the velocity in miles per hour. Making these substitu- 
tions and reducing, we have 


V2 
06688— 
es Ge as <<) enc eet at (311) 


pre 
fs 
vam +( 06688 a 
By this formula Table XIII. is calculated for the standard 
gauge g = 4' 83", = 4.708. 
An approximate formula may be obtained by assuming that 
ab = g for practicable values of ¢e. Substituting this m the 


5280 


first value of e given above, and replacing 7 by 3600 V? we 
have 

; 
{approx.) ei ogess oF (312) 


which is the formula generally employed. 


TURNOUTS. 


In laying a new track, the transverse inclination is first 
given to the ballast by grade pegs driven either side of the 
centre line at a distance of (g + .188) each side of the centre, 
the outside peg being set higher, and the inside peg lower 
than the grade of ballast on the centre line, by the proper 
elevation selected from Table XIII. But in re-surfacing an 
old track, the inner rail is taken as grade and the outer rail is 
raised the necessary amount. 


202. The proper elevation may be found mechan- 
ically by the following method : 

To find, on a curved track, the length of a chord whose middle 
ordinate shall equal the proper elevation of the outer rail for any 
velocity V in miles per hour. 

By the conditions of the problem, we have m, in eq. (809) 
equal to ¢ in eq. (312), or 
ec  gV* .06688 


SR R 
a. c= .73144 VV9 (318) 
When g = 4.708, 
¢ = 1,587V (314) 


Lay off the chord, ¢, upon the rail of the track, stretch a 
piece of twine between the points so found, and measure the 
middle ordinate; it will equal the proper elevation. 


203. The velocity assumed in the preceding formule 
should be that of the fastest regular trains which will pass 
over the curve in question, since the flanges would be forced 
against the outer rail were there no centrifugal force devel- 
oped, by reason of the wheels being rigidly attached to the 
axles, and the axles being parallel. 

The rails on tangents should be level transversely, except 
near curves, where for 50 or 100 feet from the curve one rail 
is gradually raised, so that at the P.C. or P.T. it may have 
the full elevation due to the curve. Ata P.C.@. the elevation 
should be an average of the elevations due to the two arcs. 
Owing to the difficulty of properly adjusting the elevation of, 
rail, it is objectionable to have arcs of very dissimilar radii 
join each other; and the objection is much greater in the case 
of reversed curves unless separated by a short tangent. See 


g 82, 


We 


FIELD ENGINEERING. 


On the other hand, a short tangent between arcs which 
curve in the same direction should be avoided, since it makes 
a ‘‘flat place” both in line and levels, at once unsightly and 
injurious to the rolling stock. 

In the case of turnouts, however, no elevation of rail is pos 
sible (except when both tracks curve in the same direction); 
hence reversed curves are allowable, the speed of trains being 
usually quite low also. 


204. The coning of the wheels, by which the 
wheel on the outer rail gains a diameter enough larger than 
the other to compensate for the superior length of the outer 
rail, although a theoretically perfect device, 1s gradually going 
into disuse. To be effective for the sharpest curves, the coning 
must be so great as to produce an unsteady motion on tan- 
gents, very objectionable at high speeds. Moreover, it is un- 
desirable to seek for an equilibrium of lateral forces in a car 
on a curve, since the flanges are then sure to strike the inner 
and outer rails alternately with damaging force, as that equi 
librium is momentarily disturbed. It is far better that the 
flange should press steadily against the outer rail, while that 
pressure is modified and reduced somewhat by the elevation 
of the rail. For these and other reasons, car-wheels are now 
made nearly cylindrical. 


LEVELLING. 181 


CHAPTER VIII. 
LEVELLING. 


205. The field operations with the Engincer’s Level are of 
a more simple character than those performed with the transit, 
yet require equal skill and nicety of manipulation in order to 
produce trustworthy results. The transit is used to ascertain 
the relative horizontal position of points, the level to obtain 
their relative vertical position. 


206. In order to express the elevation of points, they must 
be referred to some level surface of known (or assumed) eleva- 
tion; and in order that the elevations may all be positive up- 
ward, this surface of reference should be selected below all the 
points to be considered. The level surface of reference is called 
the datum. 

The elevation of the datum és always zero. The elevation of any 
point is its vertical height above the datum. 

Near the coast the sea level is usually adopted as the datum; 
inland, the low water mark of a river or lake, etc. ; butitis not 
necessary that the datwm should coincide with a water surface. 
If any points whose elevations are to be ascertained are below 
the water surface, the latter may be assumed to have an eleva- 
tion of 100 or 1000 feet instead of zero; that is, we remove the 
datum, in imagination, to 100 or 1000 feet below the level of 
the water surface. 


207. Incase of a survey commencing at a point quite re- 
mote from any important water surface, any permanent point 
may be selected as the original point of reference, and its ele- 
vation may be assumed at 100 or any other number of feet; 
that is, we fix the datum at the same number of feet below that 
point. The point of reference is called a bench, or bench- 
mark, and is designated by the initials B.M. Other benches 
are established at intervals during a survey, and their eleva- 
tions determined instrumentally. They are then convenicnt 


FIELD ENGINEERING. 


points of known elevation for future reference. We cannot 
assume the elevation of more than one bench on the same sur- 
vey, else we should have more than one datum, and all the 
results would be thrown into confusion. 


208. Having established the first bench and recorded its 
elevation, the next step is to set up the instrument firmly at a 
moderate distance from the bench, so that the telescope shall 
be somewhat higher than the bench, and in full view of a rod 
held vertically upon it. The instrument having been tested for 
its several adjustments, and found to be correct, the line of sight 
through the intersection of the cross-hairsis known to be hori- 
zontal when the bubble stands at the middle of its tube. Turn- 
ing the line of sight upon the rod, the point of the rod covered 
by the horizontal cross-hair is known to be on a level with the 
cross-hair; and the latter is therefore Aigher than the bench by 
the distance intercepted on the rod from its lower end. Add- 
ing this distance to the elevation of the bench, we obtain the 
clevation of the cross-hair, known technically as the ** Height 
of Instrument,’’ and designated by the initials ZZ.Z. 


209. The distance intercepted on a rod from its lower end 
by the line of sight, when the rod is held vertically on any 
given point, is called the reading of the rod at that point. 


210. Having obtained the height of instrument, the eleva- 
tion of any point somewhat lower than the cross-hair is easily 
ascertained by taking a reading of the rod upon it. The read- 
ing subtracted from the height of instrument gives the eleva- 
tion of the point above the datum. The elevation of any num- 
ber of other points may be similarly obtained. But the eleva- 
tion of points on the ground higher than the cross-hair, or 
farther below it than the length of the rod, cannot be deter- 
mined, because in either case the line of sight will not cut the 
rod, and hence there can be no reading. In order to observe 
such points, the instrument must be removed to a new posi- 
tion, higher or lower than before, as the case may require. 


211. Before the instrument is removed to a new position, 
a temporary bench, called a Turning Point (and designated 
by 7.P.or ‘‘Peg’”’) must be established, and its elevation ascer- 


LEVELLING. 183 


tained as for any other point, but with more care. A turning 
point must be a firm and definite point whose position cannot 
readily be altered in the least, nor lost sight of. A small stake 
firmly driven, or a point of rock projecting upward, is fre- 
quently used. The reading having been taken on the turning 
point, the instrument is carried forward to a new position, 
levelled up properly, and the new Height of Instrument ob- 
tained by a new reading on the same turning point. Since the 
cross-hair is higher than the point (otherwise there could be no 
reading) the reading, added to the elevation of the point, gives 
the Height of Instrument. 


212. In general, the intersection of the cross-hairs being 
higher than any point on which a reading is taken: 
0 find the Height of Instrument, add the reading on a point 
to the elevation of the point; and :, 
To find the Elevation of point, subtract the reading on at 
from the Height of Instrument. 


A reading taken for the purpose of finding the Height of 


Instrument is called a Backsight (B.S). A reading taken 
for the purpose of finding the elevation of a turning-point (or 
of a bench used as such) is called a Foresight (F.8). Hence 
Backsights are always plus, and Foresights always minus. 


213. The form of fieid-book used for the survey of 
a railroad, or other continuous line, is shown below. The jirst 
column contains the numbers of the stations on the line and 
of plus distances to other points on the line where readings are 
taken—also the initials of benches and turning points, in 
order, as they occur. The sceond column contains the back- 
sights, taken on points of known elevation only. The third 
column contains the height of instrument, recorded on the 
same line as the elevation of the turning point (or bench) from 
which it is calculated. The fourth column contains the fore- 
sights, taken on new turning points, and benches used as such, 
only. The jifth column contains the readings taken on all 
other points noted in the first column. The stzth column con- 
tains the elevations of all points observed. The right-hand 
page is reserved for remarks, descriptive of the benches and 
their location—of objects crossed by the line, as roads, streams, 
swamps, ditches, etc. ; the depths of streams, etc. 


184 FIELD ENGINEERING. 


LEVEL BOOK. 


Sta, BES: H.I. E.S. Rod. Elev. Remarks. 
B.M. | 4.683 | 204.683 200.000 | White oak, 115 R. 
0 2.1 202.6 
1 3.4 201.3 | 
+ 50 | 5.2 199.5 
Peg | 1.791 | 197.260 | 9.214 | 195.469 
pas Ligon 193.0. 
+ 25 | 7.0 190.3 Brook 5 wide; 1 deep 
+ 50 3.1 194.2 
3 0.5 196.8 
Peg | 11.750 | 208.574 | 0.436 | 196.824 
Peg | 11.938 | 219.528 | 0.979 | 207 595 
+ 90 | | 3.5 S16:01= 9) 
4 2.6 216.9 
B.M. | | 2.075 | 217.453 | Maple, 78 L. 
5 a. 217.8) | 
o | 0.9 218.6 
Peg | 9.005 | 227.801 | 0.732 218.796 
7 | 6.2 221.6 -| 
39.162 | 11.361 | 


When a bench is not used as a turning point, the reading on 
it is recorded in the fifth column. 

The numbers in the second, fourth, and fifth columns come 
directly from the rod, those in the third are obtained by 
addition, those in the sixth by subtraction, according to the 
rule given above. The additions and subtractions made on 
each page should be proved before proceeding to the calcula- 
tions of the next. When correct, the difference of the sums 
of the backsights and foresights on the page equals the differ- 
ence of the first and last elevations on the page. Thus, in the 
form given 


(39,162 — 11.361) = (227.801 — 200.000) = 27.801 


In this proof we ignore all elevations except those of turn- 
ing points, and benches used as such, and the height of instru- 
ment, 

At the end of the survey, as well as at the end of each day’s 
work, a bench is established from which the survey may be 
resumed at any future time See &§ 28, 29, and 80. 


214. The object of making such a survey with level and 
rod is to furnish a profile or vertical section of the entire 
line, showing in detail the rise and fall of the surface over 


a 


LEVELLING. 183 


which it passes. The profile is plotted on profile-paper pub. 
lished for the purpose, the horizontal scale being usually 400 
feet to an inch, and the vertical scale 30 feet to an inch. This 
distortion of scale magnifies the vertical measures so that 
slight changes in the elevation of the surface may be seen 
distinctly. 


215. When only the difference of level of two extreme 
points is required, the survey is more simple. No readings 
are taken except on turning-points, the backsights and fore- 
sights being recorded in separate columns. No calculation is 
required until the survey is finished, when—the first reading 
having been taken on one of the given points, and the last on 
the other—the difference of the sums of the backsights and 
foresights is the difference in elevation of the two points, ac- 
cording to the method of proof mentioned in § 218. Thus the 
difference in level of any two benches established on a previ- 
ous survey may be tested, and, if found correct, all the inter- 
mediate elevations on the line may be assumed to be correct 
also. The discrepancy should not exceed one tenth of a foot 
in any case, and is usually much less. 


216. Any lack of adjustment in the instrument gives 
the line of sight a slight angle of elevation or depression, 
causing a slight error in every reading, proportional to the 
distance of the rod from the instrument. But the errors being 
equal for equal distances, and the backsights and foresights 
having opposite signs in our calculations, the errors cancel 
when the distances are equal. Hence, to avoid errors in ele- 
vation, each new turnin’g-point should be as nearly as possible 
at the same distance from the instrument as the point on which 
the last backsight was taken. For precise reading, the rod 
should not be more than 400 feet from the instrument. 


217. Another cause of error in readings is want of verti- 
cality in the rod. This may be avoided by the use of a disk- 
level, or in the absence of wind, by balancing the rod. The 
rod may be plumbed one way by the vertical cross-hair of the 
level, and to ensure a vertical reading in the plane of the line of 
sight, the rod may be gently waved each side of the vertical 
toward and from the instrument, the shortest reading being 


Sppremapeiee MPF RE SIO — 
= SS ae 
ev 


2 


ae 


186 FIELD ENGINEERING. 


the correct one; or in case of a target rod, the target should 
rise to, but not above the horizontal cross-hair, as the rod is 
waved. 


218. When very long sights are required to be taken with 
the level, another source of error must be considered, namely, 
the curvature of the earth. 

A level line is parallel to a great circle of the earth, and is 
therefore an arc of a circle, or may be so considered. 

A horizontal line is a straight line parallel to the plane of the 
horizon. Therefore the line of sight, being a horizontal line, 
is tangent to the circle of a level line passing through the in- 
strument. 

To find the correction in elevation due to curvature of the 
earth for any distant station. Fig. 82. 


Fie. 82. 


Let A be the station of the instrument J, and B the distant 
station observed. 

Let &, = CI= the radius of curvature of the earth, or of the 
parallel arc JD. Let Z, = ID = the level distance between 
Aand B. Then JH, perpendicular to CZ, is the line of sight, 
BE is the reading of the rod, and DE = EH, = the correction 
due to curvature. 

By Tab. I., 24, JH? = DH (DE+ 2R.); but since DZ is 
very small compared with 2f,, it may be omitted from the 
parenthesis, and since JH = ID = L, very nearly, because 
the angle ACB is very small, we have Z,? = 2R,£.. 


2 
B= ae (815) 


Ss 


LEVELLING. 18% 


219. Refraction. In observing distant stations the 
line of sight passing through the atmosphere is refracted from 
the straight line JH, Fig. 82, and takes the form of a curve, 
which, for practical purposes, may be considered as the arc of 
a circle, concave downwards. Its radius, depending on the 
conditions of the atmosphere, varies from 5} to 74 times the 
radius of curvature of the earth. 7R, is considered a good 
average value. 

Refraction causes the observed object to appear too high, 
while the curvature of the earth causes it to appear too low ;— a] 
the effects being contrary, the correction for curvature is re- 
duced by the correction for refraction. If we let H, = the Wi 
total correction for both curvature and refraction, to be added Hi 
to the apparent elevation of the observed object, then Hi 


H, =~E,=53° (316) 


Table XVII. is calculated by this formula, assuming a mean Hi 
value of R, = 20,918,650 feet. 


220. The form of the earth is approximately an el- 
lipsoid of revolution. Its meridian section at the mean level | 
of the sea is an ellipse, the semi-axes of which are, according i 
to Clarke, 

at the equator A = 6378206 metres [6.8046985] | 
at the poles B= 6356584 ‘“‘ — [6.8082288] ' 


According to the same authority 


1 metre = 3.280869 feet [0.5159889] Wy 
ae 
Therefore the semi-axes expressed in feet are | 
A = 20 926 058 feet ['7.8206874] , | 
= 20855119 “ ['7.3192127 


Then the radius of curvature of the meridian 


9 


at the equator, es = R, = 20 784 422 ft. [7.3177379) 


A2 
at the poles, a = R, = 20997 240 ‘ ['7.3221622] 


188 FIELD ENGINEERING. 


In latitude 40° the radius of curvature of the meridian is 
20 871 900, and of a section at right angles to the meridian, 
20 955 400; the mean valuc,.or 22, = 20 913 650 [7.820430], be- 
ing adopted for general use. The error in the correction H, 
eq. (316) due to this assumption will usually be much less than 
that due to the assumed value of the radius of refraction. 


221. Levelling by Transit or Theodolite. When 
a transit has a level-tube attached to the telescope, it may 
be used as a Theodolite for levelling, and for taking vertical 
angles. If the instrument be in perfect adjustment, the line 
of sight will be horizontal when the bubble stands at the 
middle point of the tube, and the reading of the vertical circle 
will be zero. Should there be a small reading when the line of 
sight 1s horizontal it is called the ¢rdex error. When the line 
of sight is not horizontal, the angle which it makes with the 
plane of the horizon is called an angle of elevation, or of de- 
pression, according as the object upon which the line of sight 
is directed is above or below the telescope. This angle is 
measured on the vertical circle, being the difference of the 
reading and the index error, when both are on the same side 
of the zero mark, and their swm, when they are on opposite 
sides. When the distance to an observed object is known, 
and its angle of elevation or depression is measured, we may 
calculate its vertical height above or below the telescope. 


§ elevation 
’ depression 
sb I, = the horizontal distance 


Let + a = angle of 


‘‘ I’ = the distance parallel to 
line of sight 


& h = difference in elevation of 
object and instrument. 


Then for short distances, 


hA=Ltane=WL' sina (817) 


Fie, 83. For long distances the curvature of 

the earth and refraction must be considered. Fig. 838. 
Tet.J be the place of the instrument, and F’ the object 
observed, 


S| 


LEVELLING. 189 


Let Z, = the distance, measured on the chord of the level 
arc LD, passing through the instrument; and let # = the 
number of seconds in the arc JD; hence, since for ordinary 
distances the chord and are are sensibly equal, 


b= = 206264".8 [5.314429] . 


or giving to R, its mean valué, § 220, 
wy = L, X .0098627 [7.993995] 


or a fraction less than 1” per 100 feet. 

Let IF’ be the arc of the refracted ray, and assuming that its 
radius is 72,, the arc will contain 4th the number of seconds 
of the arc LF 

IF’, tangent to IF, is the direction of the telescope; J/ is 
the chord of the are JF, and JZ is the horizontal. 


Let a — EIF' =observed angle of elevation. Then HIF = 
true angle of elevation = HIF” — fUIRP=a — + 4b a— 


.O71y. i 
The angle HID = 4) .°. DIF = 4p + a — .071yp; and Hi 

IDF = 90° + bbe FD = 90° -— We — O71y). | 
We now solve the triangle [7D for the side DF = h, and He 

find } 


sin (4% + a — .071y) (318) | 


pr L, cos (wb ae ea 0717) 


For an observed angle of depression make a negative in the 
formula. || 

The coefficient .071 is called the coefficient of refraction, this |) 
being a fair average value, while its extreme range is from .067 | 
to .100 under varying conditions of the atmosphere, and values | 
of the angle a. Wt 

When the difference in elevation of two or more distant 
objects is required, we obtain the elevation of each separately, 
and subtract one elevation from another. The elevation of the 
observed object is given by (7. 1.) + A. 


222. To find the Height of Instrument of a transit or 
theodolite by an observation of the horizon. Fig. 84. 


Sree Be OS 


190 FIELD ENGINEERING. 


Let J be the place of the instrument, and let « = observed 
angle of depression of the horizon. 

Let # be the point where the refracted ray meets the level 
surface, and draw the chords JF’ and AF. 

Let w= the angle ACF, let h = AJ, and let & = the coefii- 
cient of refraction. 

In the triangle JAP, 


IAF = 90° +44, AFT = 4 — kp, AIF = 90° —  — kp) 


Hence FIE = » — kt. But FIH=a-+ ky 


(a 


b= = (319) 


Let F” be the tangent point of a right line drawn through J; 


Fia, 84 


then AI = OF" exsec ACF", but CF" = R,, and, since 7 is 


always very small, ACH" = 4(# + @) very nearly = — a 


Ven hk 
p= Jey exsec any a (820) 


Giving to R, its mean value, § 220, and assuming k= 7; 


log h = 7.820480 + log exsec 1.0801 a (321) 


at 


LEVELLING. 19} 


Otherwise, we may solve the triangle AIF since 


sin (3 — ky) 


: Fae Fee as a sin 4@ (322) 
Be = 2h, sin 3 — 21° at - 
COS 7 5h 
When &k = ze 
sin 4a@ 
—~) it ZC 
h = 2R, sin ca. nae (823) 


Example.—The observed dip of the sea horizon is 24 =a 
What is the height of the instrument above the sea? 


By eq. (821) 1.0801 x a@ X 60 = 1555".34 3.191825 
9 


6.383650 
Table XXVI. q—22 9.070130 
R 7320430 


° 


h = 594.58 2.7'74210 


Methods of determining heights by distant observations can- 
not be relied on for more than approximate results, since they 
necessarily involve the uncertain element of refraction, and 
usually a lack of precision in the vertical angle, the are reading 
only to minutes in ordinary instruments. These methods, how- 
ever, are useful where no great accuracy is required, as for a 
temporary purpose until levels can be taken in the regular way, 
or for interpolating between points of established elevation, 


223. Stadia Measurements. 

It is sometimes convenient to determine distances by instru- 
mental observation For this purpose two additional cross- 
hairs may be placed in the telescope parallel to each other and 
equidistant from the central cross-hair. These are called stadia 
hairs, and distances determined hy them are called stadia 
measurements. The stadia hairs are adjusted so as to inter- 
cept a certain space on a rod held at a certain distance from 
the instrument and perpendicular to the line of sight. For any 


192 FIELD ENGINEERING. 


other place of the rod, the distances and intercepted spaces 
are nearly proportional. The exact relation is given below. 
Fig. 88. 

Let / = AB, the distance of the rod from the vertical axis 
of the instrument; ¢ = the distance from the axis to the ob: 
ject glass of the telescope; 4 = the distance from the object- 


Fie. 85. 


glass to the rod; ¢ = the space between the stadia hairs; s = 
CD the space intercepted by them on the rod; and f = the 
focal distance of the object-glass. We then have by optics, 


Ss a— : . 
cat oF, whence a — f as and sinceea = 1l—c.°. l— 
oe 


(f+ o= 78 


f, and the space between the stadia hairs ¢ are constant, while 
sand c vary with 7. For any other distance lJ’, we then have 


Now in any given instrument the focal distance 


li—(fte)= Ly, and combining the two equations 
. Seay 
l= (ft =< +e] (324) 


s'is usually assumed at 1 foot and /' — (f+ c’) at 100 feet. 
and the stadia hairs are then adjusted accordingly. The focal 
distance f may be found by removing the object glass and ex- 
posing it to the rays of the sun and noting at what distance 
from the surface of the lens the rays form a perfect and min. 
ute image of the sun on a smooth surface; the distance ¢’ is 
measured on the telescope when the rod is clearly in focus, 
at the assumed distance. 

To measure any other distance, the rod is again observed 
at the desired point, and the space s noted, which, placed in 
eq. (824), gives 1 — (f+ ce) at once. We then measure ¢ on 
the telescope, and adding (f+ c), obtain /, the distance re- 
quired, 


LEVELLING. 193 


But inasmuch as ¢ has but a small range of values, it will 
usually be sufficient to assume for it a mean value, as a con- 
stant. In this case we may find the value of (f+ ¢) = 1 
for the instrument used. Making c’ = ¢ in eq, (824), and solv- 
ing for (f+ c), we have 

2 sl’ — s'l 
Tees ae a 
8 


— 


iv) 


=~ 
iS) 
ve 
CU 
—— 


and by laying off on level ground any two distances from the 
instrument for 7’ and J, as 100 and 500, and observing the 
corresponding spaces s’ and s intercepted on a rod, we insert 
them in eq. (825) and find (f+ o). 

Having found (f+ ¢), lay off (100 + f-+ ¢) from the instru- 
ment and adjust the stadia hairs to inclose just one foot on 
the rod at that distance. Any other distance is then found by 


the formula, 
$= 100s+(f+9 (326) 


Exam ple.—At l’ = 100 we finds’ =: 1.00, and at = 500 we 
find s = 5.061. 


. 506.1 — 500 
(QD eat re mete — 9 
Hence, eq. (825) ft+e= 061 = 1.502 


and eq. (326) 1 =100s-+-1.5; provided the stadia hairs be ad- 
justed so as to intercept 1 foot at 101.5 fect distance from the 
centre of the instrument. 


224. The foregoing formule are all that are necessary for 
horizontal sights, but since the line of collimation is generally 
inclined more or less to the horizon, 1t follows that the stadia 
hairs will intercept a larger space on the vertical rod than 
that due to the true horizontal distance. We therefore require 
a formula for reducing inclined measurements 
to the horizontal. Fig. 86. 


Let « = HFG = the angle of inclination of the line of colli- 
mation JG; 
“ §@ = CFD = the visual angle defined by the stadia hairs; 
«< ¢ — OD = space intercepted on a vertical rod. 


Then (Fig. 85), 
CGH 3 
re if — 2 face re ee ns — =F 5! 
tan 34 EF * er Pr) 


a. 
(SO) 
© 

ko) 


194 FIELD ENGINEERING. 
Th In Fig. 86 
' s= CE — DE = EF [tan (a + 49) — tan (a — 46)] 
while the true value (for the same distance) would be 
C'D' = 2EF tan 48 
Dividing one by the other we derive 


CD 2 tan 40 a 
$. tan (a + 46) — tan (a — 46) 


By giving to 8s’ and /’—(f+ ec) in eq. (827) their customary 


Fig. 86. 


values, v7z., 1 and 100, we have tan46=.005 .:°. 6 = 34’ 22’.63 
and by Trig. Table II. 70, 


sin 9 
aH tan (v@-+-46) — tan (a — 46) = ———_____ —_ 

iia Coase ot ( ) cos (aw + 49) cos (~—44) 
Since 6 is small, we have sensibly 

Hoi sin 6 = 2 tan 49, and cos (@ + 49) cos (a — 49) = cos? « 

Hii | and the last equation reduces sensibly to 

it C'D 

Hi rae cos? @ (328) 

/ 8 

which is the coefficient of reduction required by which 
i to multiply the observed space s in case of inclined sights. 


Hence the formula for distance (eq. 326) becomes in this case 
without sensible error 


{| t= 100 s cos? a+ (f+e) (829) 


Tables XVIII. and XIX. have been calculated by the exact 
formula for the coefficient. 


LEVELLING. 19d 


Example.--Given : « = 8° 20' and s = 9,221; what is the 
horizontal distance to the rod? 
Eq. (829) 100 log...2. 
8 9.221 0.964778 | 
a 8° 20' Tab. XTX. «¢ 9.99078) 
902.7 2.955558 
+e 1.5 .. Ans. 904.2 ft. 


The rodman should have a disk level to insure keeping the 
ro‘ vertical. 


225. Another method of procedure is that in which 
the rod is always held perpendicular to the line of collimation, 
however much inclined the latter may be. To secure this posi- 
tion of the rod, a small brass bar is attached, having sights 
upon it through which the rodman watches the instrument 
during an observation, the line of sight being at right angles to 
the rod. The distance thus obtained is of course parallel to 
the linc of collimation, and requires to be reduced to the hori- 
zontal. 

For this purpose, we have (Fig. 87). 


Fia. 87. 
TH=IGcosa+ LG sin a 
or IH = (100 s+ f+e)cosa+r sin a (330) 


in which r is the reading of the rod by the line of collimation. 
For the elevation of the point B above J, Hy 


EB = HG — GBcosa 
or ERB = (100s+ f+ ¢) sin «~— 7 cos a (3831) 


196 FIELD ENGINEERING. 


When the distances are sufficiently great, correction must be 
made for curvature of the earth and refraction, as already ex- 
plained. 

This method is employed by the topographical parties of the 
U. §. Coast Survey in connection with the plane table. Their 
instruments, however, are so constructed as to give distances 
in metres, and heights in feet, requiring a modification of the 
above formule. 


OMAP ih ix. 
CONSTRUCTION. 


HH 226. The engineer department of a railway com- 
| pany is usually reorganized for the construction of the road, 
as follows: Chief engineer, Division engineers, Resident 
engineers, Assistant engineers. On some roads the division 
engineers are styled ‘‘ Principal Assistants ;’ the resident 
engineers, ‘‘ Assistants;” and the assistant engineers are de- 
! signated according to their duties, as ‘‘leveller,” ‘ rodman,” 
Hi etc. | 

A resident engineer has charge of‘a few miles of line, 

i limited to so much as he can personalfy superintend and 

direct. He has one or more assistants and an axman in his 
i party. All instrumental work is done and all measurements 
taken by the resident engineer and his assistants. 

A division engineer has charge of several residencies, 
| and inspects the progress of the work on his division once 
\'( | or twice a week. In his office, which should be centrally 
| located, all maps, profiles, plans, and most of the working 
bial drawings required on his division are prepared. To him the 
HGH resident engineers make detailed, reports once a month, or 

| oftener if necessary, which he passes upon as to their cor- 
rectness, and from which he makes up a monthly report, or 
estimate, of the amount and value of the work done and ma- 
terials provided by each contractor on his division. The esti- 
mates are forwarded about the first of each month to the 
chief engineer, who examines and approves them, returning 
for modification any that seem to require it. 


9 


CONSTRUCTION. 19% 


The chief engineer has charge of the entire work, 
and directs the general business of the engineer department. 
He occasionally inspects the work along the line. 


227. Clearing and Grubbing. The first step in 
the work of construction is to clear off all growth of timber 
within the limits of the right of way. The resident engineer 
with his party passes over the line, making offsets to the right 
and left, and blazing the trees which stand on, or just within, 
the limits of the company’s property. The blazed spot is 
marked with a letter 0, as a guide to the contractor. After 
felling, the valuable timber should* be piled near the boun- 
dury lines, to be saved as the property of the company. The 
brushwood 1s burned. 

Where a deep cut is to be made, the stumps are left to be 
removed as the earth is excavated. In very shallow cuts and 
fills the contractor will generally prefer to tear up the trees 
by their roots’ at once, rather than to grub out the stumps 
after clearing. Where the embankments will be over three 
feet high, grubbing is not necessary; but the trees require to 
pe low-chopped, leaving no stump above the roots. The engi- 
neer should indicate to the contractor the localities where each 


process is suitable. 


clearing is in progress, the engineer should 
run a line of test Iqvels touching on all the benches to verify 
their elevations ; Ife may also rerun the centre line, replacing 
any stakes that may have disappeared, and setting guard plugs 
to any important transit points which may not have been 
previously guarded. If any changes in the alignment have 
been ordered, these may be made at the same time. 


228. While th 


229. Cross Sections. The resident engineer is fur- 
nished with ‘a profile of the portion of the line in his charge, 
upon which 1s plainly indicated by line and figures the estab- 
lished grade, From this he calculates the elevation of grade 
at each station, and by subtracting this from the elevation of 
the surface, he derives the depth of cut or fill (+ or —) to be 
made at each point. The grade given on the profile is that 
which is subsequently called the subgrade, being the surface 
of the road-bed. The final or true grade is the upper surface 
of the ties after the track is laid. 


198 FIELD ENGINEERING. 


The base of a cross section is identical with the width of the 
road-bed. It is made wider in cuts than in fills to allow for 
the side ditches. Six feet should be allowed in earth, and 
four feet in rock cuts. The ratio of the side slopes 
depends upon the material. The usual slope ratio for earth is 
1} horizontal to 1 vertical for both excavation and embank- 
ment. Damp clay and solid gravel beds will stand for a time 
in cuts at 1 to 1, or an angle of 45°, but this cannot be perma- 
nently depended on. On the other hand, fine sand and very 
wet clay may require slopes of 12 to 1 or 2 to 1. Exceptional 
cases require slopes of 3 or 4 to 1. In rock work the slopes are 
usually made at } to 1 for solid, 4 to 1 for loose, and 1 tol for 
very loose rock, liable to Tone ceed Rock embankmenis 
stand at 1 to 1. 


230. All cross sections are taken in vertical planes at 
right angles to the direction of the centre line. Figs. 88, 89. 
Formule. 


Let 6 = AB, the base of section, or road-bed. 


Rtas ad # WA Spent % We 
2 Diy aca the slope ratio 


““ d = CG = the cut (or fill) at the centre stake. 
“ h = DH or HN = the cut (or fill) at the side stake. 


x = CD = the ‘‘distance out”’ from centre to side stake. 
eta teem Cica kD), 


We have at once from the figures the general formula 
x= 4b-+s8h (3832) 
When the ground is level transversely, 
h=d, and # = 40-4 sd, 


For embankment use the same formula, considering d or h as 
positive in this case also, the figure being simply inverted. 


"Len the ground ts inclined transven ’sely; 
h=0G+DK=d+y onthe upper side in cuts; 
w= 4b-+-sd-+ sy (333) 


and h=EN=d—y onthe lower side in cuts 


= 46+ sd — sy (334) 


CONSTRUCTION. 199 


For embankments use the same formule, but apply eq. (838) to 
the lower side and eq. (334) to the upper side, the figure being 
inverted. The points D and / on the ground are usually found 
by trial, such that the corresponding values of # and y will 
verify the formule. 

When the natural slope FD or LE is uniform its ratio 8’ may 
be found by measuring along the section the horizontal dis- 
tance necessary to change the reading of the rod 1 foot (or half 
the distance necessary to change it 2 feet, etc.). Then, having 
found the depths of cut (or fill) at /’and ZL, distant 4d from the 
centre C, we have 

BH = sh = s(h — BF) 
and AN = sh = 8 (AL — h) 
Yrom these we have, for the upper side in cuts, and lower gide 
in fills. 
' 


SRR ean 


=) 


Vise gees) BF. ..2-= 4) 4 - 


S$. te 


also, for the lower side in cuts, and upper side in fills, 


PTs Fs Be ee Beale awe AL (886) 
We also have 
h—- BF= 7 — BR | 
vag + (337) 
AL —h= eee AL 
aereet 


whence the points D and # may be found by the level. 

But points D and # thus calculated should have their post- 
tions verified by the general formula, eq. (332), lest the slope 
s may not have been perfectly uniform. 

When the natural surface intersects the base between the 
points 4A and B, the section is said to be in side hil! work, 
Fig. 90. Both portions of the section are then determined by 
eq. (833), or where the slope s' 1s reguiar, by eq. (835) measuring 
in every case from the centre stake C; but observing that 
when the centre is in cut and one side in fill, or vice versa, that 
J must be considered negative for that side, wheuce eq. (883) 
becomes for this case 

z= 4b—sdt+ ey (833) 


200 FIELD ENGINEERING. 


231. Staking out Earthwork. Beginning at a 
point on the centre line where the grade cuts the natural sur- 
face, the engineer drives a grade stake (marked 0.0) and notes 
the point in the cross-section book. If the line of intersection 
of the road-bed and surface would make an acute angle with 
the centre line, he also finds the points where the edges of the 
proposed road-bed will intersect the surface, drives grade 
stakes, and also stakes out a cross section through each of 
those points, if necessary. 

Then advancing to the next point on the centre line where 
a section is required, he finds its elevation with the level (veri- 
fying or correcting the elevation taken on the location), calcu- 
lates the depth of cut or fill CG, which 1s then marked upon 
the back of a stake there driven; a cut being designated by C 
and a jill by L. 

Lf the ground is level transversely (Fig. 88), he calculates z by 


Fig. 88. 


eq. (382) and lays off this distance at right angles to the centre 
line, driving slope stakes at the points D and #, marked with 
the depth of cut or fili. The marked side of slope stakes should 
face the centre line. 


Lf the ground ws inclined transversely (Fig. 89), he first measures 


Fie. 89. 


the distance, 45, to F’ and finds the depth BF’ for record. He 
then proceeds to find the point D. If the natural slope be uni- 
form, D may be found by eq. (335) or (337), verifying the result 
by eq. (832). The point Hof the other slope may be found 
similarly, using eq. (336) or eq. (337): verifying by eq. (332). 


CONSTRUCTION. 201 


232. If the ground be irregular, the depth of cut or fill is 
found not only at the centre and edges of the road-bed, but 
also at every other point along the cross section where the sur- 
Jace slope changes, all of which depths are recorded, together 
with their respective distances from the centre. To find the 
point D: assume a point supposed to be near D, and there 
take a reading of the rod. The difference of the readings at 
that point and at Cequals y’ for that point, which inserted in 
eq. (333) gives a value 2’, If 2’ agrees with the horizontal dis- 
tance of the assumed point from (C, the true position of D has 
been found. If 2’ be greater than this, by subtracting the eq. 

‘= 30-+ sd-+sy’' from eq. (888) we derive 


a= x + sy — ¥) (338) 


the last term of which shows the correction to be added to 2’. 
Now in advancing from the assumed point to the extremity of 

, the rise of the surface 1s approximately (y — y’), and if, in 
going the additional distance, s(y — y'), a further rise is en- 
countered, this last, multiplied by s, must also be added to 2’, 
and so on until the additional advance makes no change in the 
value of y. The point thus found, verified by eq. (882), is the 
point D required. 

But if 2’ be less than the distance of the assumed point from 
C, we have 


v= a' — y' — Y) (338)' 


the corrections being subtractive. 

The point # on the other slope is found in a similar manner, 
using eq. (834) for the value of 2’; if 2’ be greater than the as- 
sumed distance, we have 


2=e' —y—y) (339) 


the corrections being subtractive ; but if @' be less than the as. 
sumed distance, 


=v + sy —Y) (339)’ 


the corrections being additive. 

233. In side-hill work (Fig. 90) proceed in the same 
manner, using eqs, (833) or (333)' and (388) in all cases of un- 
even ground. When the surface slope s’ is uniform, eq. (335) 
may be used, if preferred, on either side. In addition to the 


ia] 


re 


eee 


202 FIELD ENGINEERING. 


centre and side stakes, a grade stake is driven at the point 0, 
where the surface intersects the grade, the stake facing down 
hill. 

To find a grade point, set the target to a reading equal to the 
height of instrument less the elevation of grade, and stand the 
rod at various points along the given line until the target coin- 
cides with the line of collimation. 


Fie. 90. 


234. When two materials are found in the same section, 
as rock overlaid with earth, each material requires its own 
slope, and a compound section is the result. To stake 
out work of this description, the depth of earth to the rock must 
be known, and may be nearly ascertained by reference to an 
adjacent section already excavated. Fig. 91. 


heal ke Ta ono a ial 


- 4 


Fria. 91. 


Let a; be the depth of earth at C 


“6 is Ct Sha 6c “6 P or Q 
‘* s, be the ratio of rock slope 
Sh Sig oe ** earth slope 
Then w= 40+ si(d — a) + 91) + 8e(de + Yo) (840) 


in which y; = difference of rod readings on the rock at C, and 
D,, or C, and #,; and y, = difference of rod readings on the 
surface at Pand Dz, or at @ and #,. The upper sign applies 
to the upper side, the lower sign to the lower, 


CONSTRUCTION. 203 


It is better, however, to make an indefinite cross profile at 
first, driving two reference stakes quite beyond the section 
limits: and when the contractor has removed the earth from 
between D, and Z,, indicate to him those exact points by 
marks on the rock, and also set the slope stakes at D. and £. 


235. The frequency with which cross sections should 
be takén depends entirely upon the form of the surface; where 
this is regular, a section at each station is sufficient. A cross 
section should be taken, not only at every point on the centre 
line where there is an angle in the profile, but also wherever 
an angle would be found in the profile of a line joining a series 
of slope stakes on either side, even though the profile of the 
centre line may be quite regular at the corresponding point :— 
the object being, not only to indicate the proper outlines of 
the earthwork, but to furnish the data necessary to calculate 
correctly the quantities of material removed. Rockwork will 
generally require more frequent sections than earthwork. 


236. Vertical Curves.—The grades as given on the 
profile are right lines, which intersect each other with angles 
more or less abrupt. ‘These angles require to be replaced by 
vertical curves, slightly changing the grade at and near the 
point of intersection. A vertical curve rarely need extend 
more than 200 feet each way from that point. Fig. 92. 


Fig. 92. 


Let AB, BOC, be two grades in profile, intersecting at station 
B, and let A and C be the adjacent stations. It is required to 
join the grades by a vertical curve extending from A to @. tt 
Suppose a chord drawn from A to (;—the elevation of the 
middle point of the chord will be a mean of the elevations of 
grade at A and (C; and one half of the difference between this 


owe 


204 FIELD ENGINEERING. 


and the elevation of grade at B will be the middle ordinate of 
the curve. Hence we have 


grade A + grade 0 | 


9 — grade B) (341) 


M=+%4 
in which M = the correction in grade for the point B. The 
correction for any other point is proportional to the square of 
its distance from Aor 0. Thus the correction at A 26 is 
3M; at A+ 50 it is $M; at A+ 75 it is {M, and the same 
for corresponding points on the other side of B. The correc- 
tions in the case shown are subtractive, since M is negative. 
They are additive when M is positive, and the curve concave 
upward. 

These corrections are made at the time the cross sections 
are taken, and the corrected grades are entered in the field- 
book opposite the numbers of the respective stations. 


237. Form of Field-book.—A complete record of 
all cross-section work is kept in the cross-section book. 
On the left-hand page is recorded, in the first column, the 
numbers of the stations and other points where sections 
are taken; in the second, the elevations of those points, copied 
in part from the location level-book, but verified or corrected 
at the time the section is taken; in the third, the elevation of 
the grade for the same points; in the fourth, the width of 
base 0b; in the fifth, the slope ratios, s; and in the sixth, the 
surface ratio s’ when uniform. The right-hand page has a 
central column, in which, and opposite the number of the 
station, is recorded the centre depth of the section, marked 
+ or —, to indicate cut or fill, as the case may require. 
To the right of this are recorded the notes of that portion of 
the section which lies on the right of the centre line, as the 
line was run, and to the left, the notes of the left side. The 
distance from the centre to each point noted is recorded as 
the numerator of a fraction, and the cut or fill at the point 
as the denominator, prefixed by a-+ or — as the case may 
require. The denominator for a grade point is zero. The 
numbers of the stations should increase wp the page, as in a 
transit book, so that there may be no confusion as to the right 
and left side of the line. The several points being noted in 
order as they occur from the centre outwards, the notes far: 


CONSTRUCTION. 205 


thest from the centre of the page usually appertain to the 
slope stakes; but in case the cross profile is extended beyond 
the slope stake, the note of the latter should be surrounded by 
a circle to distinguish it. The following form is a specimen 
of a right-hand page, with the first column only of the left- 
hand page: ; 


Sta. |} Cross | Sections. 
\| | 
83 || Hea” MGIC HOGI, SHOR OF SOR SA0 __ 20 CB 56 
© | + 8.6 + 14 417.7 4 21 | +215) 420.8 +25.6 428.3 +20.4 
| 
+ 60 || 37.5 _ 10 cod Os exis dO ek 42.6 
ti 5.0, 410.1192). oe 14 Pid 7 180 1 101 7 
89 BS 2s AO ah tiie ton 6 10 ; 31.6 
2 +38 +5.4| 19.4) + 8.5 1117.6 114.4 
ies fGen Ot IO 19,2 
S| OT eer ae he 
+ 27 0 
24 er ane ar fame: 
Larue! 9.6 aigeeacs Wining 
SI 25.99 MOVBID 15 
—12.6-11.2 | —12 | 10.6 — 5.3 
eg} BBSydi A Frn Orsi hut whier 48! eaGy 
—17.6 —16.4 | —17.6 | —19.6 —19.1 —12.4 


238. In case there is a liability to land-slips, the profiles 
of cross sections should be carried beyond the slope stakes, 
on the upper side of the cut, to any distance thought neces- 
sary to reach firm ground, and stakes driven for future refer- 
ence. When a number of consecutive cross profiles are to be 
considerably extended, it is well to first run, instrumentally, 
a line parallel to the centre line, and set stakes opposite the 
stations, taking their elevations. The intermediate surface of 
the sections may then be taken with cross-section rods if more 
convenient. See $37, 


239. In case of inaccessible ground, preventing a 
regular staking out, an indefinite profile of the section may 
generally be obtained, referred to the datwm for elevation and 
to the centre line for position, which being plotted on cross- 
section paper, and the grade line and side slopes added, shows 
to scale where the slope stakes should be, 


206 FIELD ENGINEERING. 


240. Any isolated mass of rock or earth which oc- 
curs within the limits of the slope stakes, but not included in 
the regular notes, is separately measured and noted, so that 
its contents may be computed and added to the sum of the 
same material found in the cross sections. 


241. Borrow-pits.—When the excavations will not 
suffice to complete the embankments, material may be taken 
from other localities, termed borrow-pits. These should be 
staked out by the engineer and their contents calculated, 
unless the contractor is to be paid for work by embankment 
measurements. A number of cross profiles are taken of the 
original surface, and (on the same lines) of the bottom of the 
pit after it is excavated, which furnish the depth of cutting 
at each required point. Borrow-pits should be regularly ex- 
cavated, so that they may not present an unsightly appear- 
ance when abandoned. Borrow-pits may be avoided by 
widening the cut uniformly at the time it is staked out, so 
that it may furnish sufficient material; provided the material 
is suitable, the embankment accessible, and the distance not 
too great. When the excavation is in excess, the surplus ma- 
terial should be uniformly distributed by widening the adja- 
cent embankments, if possible; otherwise it 1s deposited at 
convenient places indicated by the engineer and is said to be 
wasted, 


242. Shrinkage.—In estimating the relative amounts of 
excavation and embankment required, allowance must be made 
for difference in the spaces occupied by the material before ex- 
cavation and after it is settled in embankment. The various 
earths will be more compact in embankment, rock less so. The 
difference in volume is called shrinkage in the one case, and 
trcrease in the other. 

Shrinkage in 1000 cu. yds. 


Material. -Of excavation. Of settled embkt. 
Sand:and gravel... /s.5..2. si 80 C. Yds. 87C. Yds. 
SOLA tecnsouin's face pean nuh cattle 10055 111 35 
TAU cee ratte Salen see tua ai eae 14 Sa Leh 
NVGL SULT. OS cae ea ee eee wae es 1508 Ps 200°“ 

Increase in 1000 cu, yds. 
Rock, large fragments......... 600 C. Yds, 375 C. Yds. 
«¢ “medium tragments...... TU 413“ 


‘+ gmall seine, 8 800 i 444 * 


CONSTRUCTION. 20% 


Thus, an excavation of sand and gravel measuring 1000 cubic 
yards will form only about 920 cubic yards of embankment; or 
an embankment of 1000 cubic yards will require 1087 cubic 
yards of sand or gravel measured in excavation to fill it; but will 
require only 587 cubic yards of rock excavation, the rock being 
broken into medium-sized fragments; while 1000 cubic yards 
of the latter, measured in excavation, will form 1700 cubic 
yards of embankment. 

The lineal settlement of an earth embankment will be 
about in the ratio given above, therefore the contractor should 
be instructed in setting his poles to guide him as to the height 
of grade on un earth embankment, to add 10 per cent (average) 
to the fill marked on the stakes. In rock embankments this 
is not necessary. The engineer should see that all embank- 
ments are made full width at first, out to the slope stakes, and 
by measure at or above grade, so that the whole may settle in 
a compact mass.. Additions to the width made subsequently 
are likely to slide off. 


243. The cross-section notes should be traced in ink at the 
first opportunity to secure their permanence. An office copy 
should also be made to serve in case of loss or damage to the 
original. 


244. Alteration of Line.—Inasmuch as the centre line 
at grade is the base of reference for all measurements and cal- 
culations in earthwork, any change made in it after the work 
of grading has begun should be most carefully recorded and 
explained. The centre stakes of the old line should be left 
standing until after the new line is established, so that the per- 
pendicular offset from the old line to the new, at each station, 
may be measured, as also the distance that the new station may 
be in advance of, or behind the old one. The date of the change 
should be recorded. The original cross sections are extended 
any amount requisite, the distance out being still reckoned from 
the old centre, while a marginal note states the amount by which 
the centre has been shifted. 

The difference in length of the lines will make a long or short 
station at the point of closing. The exact length of such a 
station should be recorded, so-that 1t may be observed in re- 
tracing the linc at any time, and in calculating the quantity of 


——— = 


208 FIELD ENGINEERING. 


earthwork. The original transit notes of the altered line should 
be preserved, but marked as ‘‘ abandoned,” with a reference to 
the notes of the new line on another page. 


245. Drains and Culverts.—The engineer should ex 
amine the nature and extent of each depression in the profile 
with reference to the kind of opening required for the passage 


‘of water. For small springs, and for a limited surface of rain- 


fafl, cement pipes, in sizes varying from 12 to 24 inches diame- 
ter, serve an excellent purpose as drains. These are easily laid 
down, and if properly bedded, with the earth tamped about 
them, are very permanent; but their upper surface should be 
at least 24 feet below grade. The embankment is protected at 
the upper end of the drain by a bit of vertical wall, enclosing 
the end of the pipe. If necessary, a paved gutter may lead to 
it. ; 

Where stone abounds, the bed of a dry ravine may be partly 
filled with loose stone, extending beyond the slopes a few feet, 
which will prevent the accumulation of water. 

When the flow of water is estimated to be too great for two 
lines of the largest cement pipe, or when the embankment is 
too shallow to admit them safely, a culvert is required. A 
pavement is Jaid one foot thick, protected by a curb of stone 
or wood 8 fect deep at each end, and wide enough to allow the 
walls to be builtupon it. Itshould have a uniform slope, usu- 
ally between the hmits of 50 to 1 and 100 to 1 to ensure the 
ready flow of water. In firm soils the foundation pit is exca- 
vated one foot below the bed of the stream, butif mud is found 
this must be removed and the space filled with riprap, the up- 
per course of which is arranged to form the pavement at the 
proper level. In a V-shaped ravine, requiring too much ex- 
cavation at the sides, and where the fall 1s considerable, riprap 
may be used to advantage, the bed of the stream above the 
culvert being graded up by the same material to meet the pave- 
ment. In some cases a curtaim, or cross wall, 1s necessary on 
the lower end to retain the riprap. 

Culverts should be laid out at right angles to the centre line 
whenever practicable, the bed of the stream being altered if 
necessary. The length of an open culvert 1s the entire distance 
between slope stakes, the walls being parallel throughout, or 
the length may be taken somewhat less than this, and the walls 


? CONSTRUCTION. 2092 


turned at right angles on the upper end, forming a facing to 
the foot of the slope. The walls are carried up to grade for 
the width of the road-bed, and are stepped down to suit the 
siopes. A course is afterwards added to retain the ballast. 

In box culverts the span varies from 2 to 5 feet, the height 
in the clear from 2 to 6 feet; the thickness of walls from.3 to 
4 feet; the thickness of cover from 12 to 18 inches, and its 
length at least 2 feet greater than the span. The walls terminate 
in short head-walls built parallel to the centre line, the top 
course being a continuation of the cover. The length of a 
head-wall, measured on the outer face, 1s equal to the height of 
the culvert in the clear multiplied by the slope ratio of the 
embankment. The perpendicular distance from the centre 
line to the face of a hvad-wall 1s equal to one half the road-bed, 
plus the depth of the top of the wall below grade multiplied by 
the slope ratio, or 40 + sk. A coping 1s sometimes added. 


246. Arch culverts are used when the span required is 
more than 5 feet, and the embankment too high to warrant 
carrying the walls up to grade as an open culvert. The span 
varies from 6 to 20 feet; the arch is a semicircle, the thickness 
varying from 10 or 12 inches to 18 or 20 inches. The height 
of abutments to the springing line varies. from 2 to 10 feet, the 
thickness at the springing line from 8 to 5 feet, and at the base 
from 3 to6 feet, the back of the abutment receiving the batter. 
The foundations are laid broader and deeper than in box cul- 
verts, each abutment having its own pit, carried to any depth 
found necessary. The half length of the culvert is 40 + sh, in 
which ¥# is the depth of the crown of the arch below grade. 
The abutments are carried up half way from the spring to the 
level of the crown of the arch, and thence sloped off toward 
the crown. ‘The face walls are carried up to the crown, and 
coped. The wing walls stand at an angle of 30° with the 
axis of the culvert, they receive a batter on the face, and are 
stepped (or sloped) down to suit the embankment. Their 
thickness, at the base, 1s the same as that of the abutment; at 
the outer end 3 feet. They stop about 3 feet short of the foot 
of the slope. They need not be curved in plan. 

Any stone structure of dimensions greater than those given 
above, scarcely comes under the head of culverts, and should 
be made the subject of a special design by the engineer. 


210 FIELD ENGINEERING. 


247. Staking out Foundation Pits.—For box 
culverts.—The engineer having decided upon the size of cul- 
vert required, makes a diagram of it in plan, on a page of his 
masonry book, recording all the dimensions, stating the sta- 
tion and plus at which its centre is taken, the span and height 
of the opening, etc. He then sets the transit at the centre A, 


Fig. 93, measures the angle between the centre line and axis, 


Fie. 93. 


(making it 90° if practicable); on the axis he lays off the dis- 
tances to the ends of the culvert and drives stakes at Band C. 
Perpendicular to BC he Jays off the half widths of the pit, set- 
ting stakes at D and #, and laying off DP'and HH = AB; and 
DGand HI = AC. On IG produced he lays off CJ = CK, and 
perpendicular to this /M and AZ, and finds the intersections 
Oand V. A stake is driven at each angle, and upon it is 
marked the cut required to reach the assumed level for the 
foundation. These cuts are recorded on the corresponding 
angles of the diagram. The pit 1s thus no larger than the 
plan of the proposed masonry, and the sides are vertical, which 
answers the purpose for shallow pits. 


Kor arch culverts.—The pit for each abutment when 
shallow may be of the same dimensions as the lower founda- 
tion course . if more than five feet deep, it should be enlarged 
by an extra space of one foot all around. In Fig. 94 the inside 


CONSTRUCTION. 211 


lines show the plan of the abutments at the neat-lines ; the 
outside lines represent the pits. Having prepared a plan of 
the structure suited to the locality, and made a diagram of 
the same in the masonry book; set the transit at A, and drive 
stakes at D, H, NV and O on the centre line. Then turning to 
the axis BC, lay off AC, and set stakes at Gand J. With G 
as a centre, and a radius equal to 2D#, describe on the ground 


Fie. 94, 


an arc cutting #7 in X or UX = DE. cot 30°) may be calcu- 
lated; and on XG produced lay off GX, and perpendicular to 
this, AL. From WN lay off WP, parallel to AC, and measure 
PL asacheck. Drive a stake at each angle, marked with the 
proper cutting, and record the same on the diagram. The 
locality may require the wings to be of different lengths and 
angles, of which the engineer will judge.. Guard-plugs should 
be driven in line with the intended face of one or both abut- 
ments, so that the neat-lines can be readily given when re 
quired. In case the material 1s not likely to stand vertically, 
the pit must be staked out with sloping sides, as described 
below. 

For bridge abutments.—A design for every impor- 
tant structure is usually prepared in the office after a survey 
of the site. The foundation pit is then laid out from dimen- 
sions furnished on a tracing, but a diagram of the pit should be 
made in the masonry book as usual. When the bredge ison a tan- 
gent, Fig. 95, set the transit at A on the centre line at its inter- 
section with the axis BC of the abutment at the level of the scat. 


pate FIELD ENGINEERING. 


Deflect from the tangent the angle giving the direction of BC, 
and lay off AC, AB, setting plugs at B and C, and reference 
plugs (two on each side) on BC produced. After staking out 
ihe sides of the pit parallel to BC, set the transit at C, and 
deflect the angle for the wing, laying off CD, and driving 
stakes at the corners # and #. Two reference points are 
then set on the line CD produced. The other wing being 


Fie. 95. 


staked out in the same manner, the cut is found at each stake 
and marked and recorded. Cross sections are then taken near 
each corner, perpendicular to each side, and slope stakes 
(marked ‘‘ slope”) are driven where the slope runs out. Inter- 
mediate sections are taken when the unevenness of the ground 
makes it necessary, and the lines joining the slope stakes are 
produced to intersect, and other stakes are driven at the inter- 
sections. ‘The position of each stake is shown on the diagram, 
and the cut recorded. 

A slope of 1 to 1 is usually sufficient for pits. Ifthe material 
will not stand at 14 to 1, or if space cannot be spared for the 
slope, the sides may be carried down vertically, supported by 
sheet piling braced from within. 

The reference points should be so chosen that the points A, 
Band C may be found by intersection, on any course of the 
masonry, during the progress of construction. 

When the bridge is on a curve, the bridge-chord 
should be found and the abutments laid out from this. Fig. 96. 
The bridge-chord is a line AB, midway between the chord of 
the curve OD, joining the centres of the abutments, and a tan- 
gent to the curve at the middle point of the span. Hence 


CONSTRUCTION. 


CA = DB=+%4MN, which may be laid off, and A and B are 
the true centres of the abutments, from which the foundations 
are staked out as before. 

The distance CH = DF to the points where the bridge-chord 
cuts the curve is 0.147CD. 

Should an abutment site on a curve be inaccessible, as when. 


Fie. 96. 


under water, from any transit point P on the curve lay off. PX 
perpendicular to the tangent at MW, observing that 


PX = MQ — AC= KR (wers PM — 4 vers CM) 
and AX = PQ —tAB= R(sin PM — 4CD) 


The point A may then be found by intersection, or by direct 
measurement with a steel tape or wire, driving a long stout, 
stake to show the point above the water. Other points may 
then be approximately found, sufficient to begin operations. 

In case of a bridge of several spans, the piers are laid out in 
the same manner, from a centre point and axis. If on a curve, 
each span has its own bridge-chord, but for convenience, the 
centre of a pier may be taken on the centre line during its con- 
struction, and the bridge-chord only found for the purpose of 
placing the bridge; the piers being long enough to allow of the 
shift. 


= s 


214 FIELE ENGINEERING. 


To locate the centres of piers, a base line is re: 
quired on one or both shores, and two transits are used to give 
ihe intersections by calculated angles. When practicable the 
spans should also be measured with a steel tape or wire. 

The bed of a pit for any sort of structure should 
receive the closest scrutiny of the engineer, it being his duty 
to judge whether the material will resist une Joad to be im- 
posed upon it. A pit may require to be excavated to a greater 
depth than first ordered, while sometimes a less depth will 
answer, aS when solid rock is found. When a good material 
is reached, if any doubt exist as to its thickness, or as to the 
character of the underlying stratum, borings should be made 
or sounding rods driven down. Piles may be driven to gain 
the requisite firmness, and a layer of riprap, of beton, or of 
timber may be used to afford a uniform bearing. When satis- 
fied of the stability of the bed, the engineer finds the original 
centres, and gives points for the courses of masonry. A com- 
plete record is kept of the amount and kind of excavation, the 
materials uscd in foundation under the,masonry, and of the 
size and thickness of each foundation course of masonry; the 
notes should be taken at the time the work is done, it being 
generally impossible to take measurements thereafter. 


248. Cattle-guards are shallow pits placed at right, 
angles across the road at the fence lines to prevent the passage 
of cattle. They are either entirely open, in which case they 
should be at least 4 feet deep, or they are covered in part with 
wooden rails laid a few inches apart. The open guard is 
preferred. It is built like an open culvert except that no 
pavement is required. The stringers carrying the rails over 
any opening should be no longer than the span plus the thick- 
ness of the walls. 


249. Trestle Work.—No wooden culverts should ever 
beused. If stone cannot be had at first, two trestle bents may 
be erected, ,eaving between them a space sufficient to contain 
the stone structure to be built when the material for it can be 
brought by rail. The bents may be backed by plank to retain 
the embankment, and the stringers are then notched down an 
inch on the caps to receive the pressure of the earth, and 
render the bents mutually sustaining. The sills are prevented 
from yielding to the pressure cf the earth by being sunk in 


CONSTRUCTION. pple 


a trench, or by sheet piling. Should the span be too long, a 
central bent may be used, so as not to interfere with building 
the wall. Sometimes pile-bents may be used with greater ad- 
vantage, the piles being driven in rows of four each, and cap- 
ped to receive the stringers. In districts where suitable stone 
is entirely wanting, pile or trestle abutments and piers are 
used for the support of bridges, the piles or posts being 
arranged in groups and capped to receive the direct weight of 
the trusses. Thcy should not sustain the embankment, but 
should be connected with it by a short trestle work. 

Trestle work is frequently used as a substitute for embank- 
ment, either to lessen the first cost, or to hasten the completion 
of the line, or for lack of suitable material with which to form 
an embankment. The cost of trestle work, however, is not 
iess than that of an earth embankment formed from borrow 
pits, unless its height exceeds about 15 fect, depending on the 
relative prices of materials and labor. When not exceeding 30 
fect in height, the dents, for single track, are usually composed 
of two posts, a Cap and sill, cach 12 X 12, and two batter posts, 
10 < 12, inclined at 4th to 1, all framed together. Two lengths 
of 3-inch plank are spiked on diagonally on opposite sides of 
the bent as braces. The length of the caps should equal the 
width of the embankment; the posts should be 5 fect from 
centre to centre, and the batter posts 2 feet from the posts at 
thecap. The sill should extend about two fect beyond the 
foot of the batter post. A masonry foundation for the bent is 
preferable, though pile foundations are not uncommon, and 
some temporary structures are placed directly on a firm soil, 
supported only by mudsills laid crosswise under the sill. The 
spans, or distance between bents, may vary from 12 to 16 feet. 
The stringers should consist of 4 pieces, 2 under each rail, 
bolted together, with packing blocks to separate them 2 or 3 
inches. Over each bent and at the centre of cach span a piece 
of thick plank about 4 feet long should be placed on edge 
between the two pair of beams to preserve the proper distance 
between them, while rods pass through the beams and strain 
them up to the ends of the plank, to increase the stability of the 
beams and prevent their buckling under a load. The string- 
ers should be able to carry safely the heavicst load without 
bracing against the posts. The bents, however, if high, must 
be braced against each other. The stringers should be con 


216 FIELD ENGINEERING. 


tinuous, the two pieces breaking joints with each other at the 
bents, to which they are firmly bolted. They may rest directly 
on the caps, or corbels may intervene. -The spans on a curve 
should be shorter than on a tangent. The ties should be 
notched down to fit the stringers closely, and guard rails, either 
wood or iron, secured to them firmly. Unless the spans are 
‘very short, horizontal bracing should be employed consisting 
of 3-inch plank, extending from the centre of each span to the 
ends of the caps, which are notched down to receive the plank. 
For trestles much higher than 80 feet the cluster bent is 
preferable, so termed because each vertical post is composed of 
a cluster of four pieces, 8 x 8, standing a little apart to allow 
the horizontal members to pass between them. The verticals 
are continuous, breaking joints, two and two, while the hori- 
zontals pass the posts and are bolted to them at the joints; the 
framing is accomplished entirely by packing blocks and bolts. 
The batter posts consist each of two pieces 8 X 8; the horizon- 
tals may be 4 X10, and extend not only across the bent, but 
Ht from one bent to another. Proper bracing is also used in every 
| direction. When very high, a secondary pair of batter posts 
may be introduced in the lower part of the structure. The 
‘| batter need not exceed 1th to1. In some instances two adjoin- 
1 ing bents are strongly braced together, forming a tower or pier, 
i and the piers placed from 50 to 100 feet apart, the roadway 
being carried on trussed bridges. The cluster bent admits of 
| any piece being removed and a new one inserted when neces- 
Ht sary. 

Iron trestles are now adopted where a permanent struc- 
ture is desired. Owing to the expansion of the metal by heat, 
the bents cannot be continuously connected with each other as 
in a wooden trestle; hence the pier form is resorted to, having 
spans varying from 380 to 150 fect, covered by trussed bridges, 
and the whole structure is more properly styled a viaduct. 


250. Tunnels. Tunnels are adopted in certain cases to 
avoid excessive excavations, steep grades, high sunimits, and 
circuitous routes. Their disadvantages are the increased time 
and cost of their construction compared with an open line, and 
their lack of light and fresh air when in use. It is desirable 
| that they should be on a tangent throughout, both for the ad- 


I mission of light and for convenience of alignment. Many 


CONSTRUCTION. Q1% 


tunnels, however, have been built with a curve at one or both 
ends.* 

The location ofa tunnel, other things being equal, should 
be such as to make not only the tunnel proper, but also its im- 
Mediate approaches by open cut as short as possible; and the 
latter should be selected so as not to be subject to overflow, 
nor liable to land slides. ‘The material to be encountered may 
frequently be determined with tolerable accuracy by a study 
of the geological formation in the vicinity, or by actual borings. 
The most favorable material for tunnelling is a homogeneous 
self-supporting rock, devoid of springs, which does not disin- 
tegrate on exposure to the atmosphere. The worst materials 
are saturated earth and quicksands. The presence of water in 
any material increases the cost considerably. 

The alignment of a tunnel is made the subject of special 
survey, after the general location is decided, and this is more 
or less elaborate according to the length of tunnel. A perma- 
nent station is established at the highest point crossed by the 
tunnel tangent, from which, if possible, monuments are set in 
each direction at points beyond the ends of the tunnel. Ii 
there are two principal summits, stations on these will define 
the tangent, which may then be produced. The monuments 
established beyond the tunnel should be sufficiently distant to 
afford a perfect backsight from the ends of the tunnel, where 
other monuments are also established. The first quality of in- 
struments only should be used, and these perfectly adjusted, 
and the observations should be repeated many times until it is 
certain that all perceptible errors are eliminated. Since the 
line of collimation will be frequently inclined to the horizon 
at a considerable angle, it is important that it should revolve 
in a vertical plane; and to secure this, a sensitive bubble tube 
should be attached to the horizontal axis, at right angles to the 
telescope of the transit. The distance may be obtained by tri- 
angulation, though direct measurement is to be preferred. A 
steel tape is convenient and accurate, providing that allowance 
be made for variations due to temperature, from an assumed 
standard. The rods described in § 48 may be uscd instead of 


*The Mont Cenis tunnel, requiring a curve at each end, was first 
opened on the tangent produced, giving a straight line through, and the 
curves were excavated subsequently. 


218 FIELD ENGINEERING. 


plumb lines, the tape being held at right angles to them, and 
therefore horizontal. A plug should be driven for each rod to 
stand on, and a centre set to indicate the line and measure- 
ment. 

As the excavation of the tunnel proceeds, the centre line is 
given at short intervals by points either on the floor or roof. 
Overhead points are generally preferred, from which short 
plumb lines may be hung, constantly indicating the line, with 
little danger of being disturbed. When a new transit point is 
required in the tunnel, it should be established directly under 
an overhead point, which serves as a check upon its perma. 
nence, and as a backsight when needed. 

Shafts are sometimes opened to give access to several points 
of the tunnel at the same time, thus facilitating the work, though 
at an increased cost. They also serve for ventilation during the 
progress of the work, though they are worse than useless for 
this purpose afterward, except possibly in the case of a single 
shaft near the centre of the tunnel. Some of the longest tun- 
nels have been formed without shafts, while many shorter ones 
have had several, which have generally been closed after the 
tunnel was completed. Shafts are either vertical, inclined, or 
nearly horizontal; in the latter case they are called adits. In: 
clined shafts should make an angle of at least 60° with the ver- 
tical. Vertical shafts may be either rectangular, round, or 
oval. Their dimensions vary, depending on their depth and 
the material encountered, between 8 and 25 feet. They are 
usually sunk on the centre line of the tunnel, though some- 
times at one side. When over the tunnel the alignment below 
is obtained directly from two plumb lines of fine wire suspended 
on opposite sides of the shaft from points very carefully deter- 
mined at the surface. The plummets are suspended in water 
to lessen their vibrations, and as soon as the transit can be set 
up at a sufficient distance to bring the lines into focus, it is 
shifted by trial into exact line with the mean of their oscilla- 
tions, the latter being very limited. Permanent points may 
then be set, but, should be repeatedly verified. As soon as the 
workings from a shaft communicate with those from either 
end, or from another shaft, the alignment thus found is 
tested, and revised if necessary. These operations require the 
ereatest nicety of observation and delicacy of manipulation to 
obtain satisfactory results. 


CONSTRUCTION, 


From plumb lines in the central shaft of the Hoosac tunnel, 
the line was produced three tenths of a mile, and met the line 
produced 2.1 miles from the west end with an error in offset 
of five sixteenths of an inch. In the Mont Cenis tunnel the 
lines met from opposite ends with ‘‘no appreciable” crror in 
alignment, while the error in measurement was about 45 feet 
in a total length of 7.6 miles. 

When a curve occurs in a tunnel it is usually near one 
end. The tunnel tangent is produced and established as 
before described, anda second tangent from some point on the 
curve outside the tunnel is produced to intersect it, the inter- 
section being precisely determined and the angle measured 
with many repetitions. The tangent distances are then calcu- 
lated, and the position of the tangent points corrected by 
precise measurements, and permanent monuments are estab- 
lished. As the tunnel advances, points may be set at short 
intervals on the curve in the usual manner; but at intervals 
of 100 feet the regular stations should be defined with finely 
centred monuments, using a 100-foot steel tape carefully sup- 
ported in a horizontal position. When it is necessary to use a 
subchord, its exact length should be calculated as shown in 
$107. When the curve has advanced so far as to render anew 
transit point necessary, this should be established at a full 
station. The subtangents from the two transit points should 
then be produced to intersect, and measured for equality with 
each other and with their calculated length. The distance 
from their intersection to the middle of the long chord should 
also be measured as a check on the deflections. When no 
perceptible errors remain, the curve may be produced as 
before ‘until the P. 7. is reached. It is evident that correct 
measure is indispensable to correct alignment on curves. 
Should obstacles on the surface necessitate triangulation, more 
than ordinary care must be exercised, and as many checks 
introduced as possible. The triangles should be so arranged 
that all of the angles and most of the sides may be measured. 

Test levels are carried over the surface with great care, 
each turning point being made a permanent bench, and its 
elevation determined with a probable error not exceeding 
0.005 foot. Levels may be carried down a shaft on a series of 
bolts or spikes abeat 12 fect apart in the same vertical line, 
the distances being measured by the same level-rod as that 


220 FIELD ENGINEERING. 


with which the benches are determined. The measures should 
be taken between two graduations of the rod, not using the 
end of the rod, which may be slightly worn. Fine horizontal 
lines on the heads of the bolts may be used to mark the exact 
distances. After the shaft reaches the level of the tunnel, the 
depth may be measured more directly with a steel tape, the 
entire length of which has been corrected at the given tem- 
perature, by comparison with the same rod. 

If the grade of a tunnel is to be continuous, it should 
be assumed at something less than the maximum of the road, 
but not less than 0.10 per station, which is required for 
drainage. If a summit is to be made in the tunnel, the grade 
from the upper end should not exceed 0.10 per station. 
Grades are given in the tunnel from day to day, or as often as 
required by the progress of the work, the marks being made 
on the sides at some arbitrary distance above grade. Turning 
points should be taken on permanent benches. 

The least width of a tunnel in the clear should be, for 
single track about 15 feet, and for double track 26 feet. The 
least height in the clear above the tie should be 18.5 feet 
for single track, and 16.5 feet at the outside rails for double 
track, allowing for tie and ballast; the roof at the centre of the 
section should be at least 20 feet above subgrade, and with a 
full centred arch 22 or 23 feet for double track. The form 
of section depends somewhat on the material traversed. In 
perfectly solid rock a nearly rectangular section may be used, 
the roof being slightly rounded. In dry clay, and stratified 
rock, a flat arch may be used, and in other cases a full-centred 
arch. .The latter form is rather to be preferred on account of 
the better ventilation afforded. The sides are made vertical, 
battered or curved, as necessity or taste may dictate. In wet 
and infirm soil an invert floor may be required, otherwise it 
is made level transversely. When a lining is required the 
original section must of course be made large enough to 
allow for the masonry, and the temporary timber supports 
behind it. Hard burned brick is usually adopted for arching, 
being durable and easily handled. In loose rock the arching 
may- be from 13 to 26 inches thick, in wet and yielding soil a 
thickness of from 26 to 89 inches may be necessary. The 
walls may be from 24 to 6 fect thick. 

In forming a tunnel, @ heading or gallery of smaller 


CONSTRUCTION. rae | 


cross section is first driven and afterwards enlarged to the 
full size required. In firm clay or loose rock which will tem- 
porarily support itself until the masonry can be put in, it is 
better to drive the heading along the floor (at subgrade) of the 
tunnel, the remaining material being then easily thrown down 
in sections as the archiny is advanced. In solid rock, or wet 
earth, a top-heading (along the roof) is generally preferred, 
The dimensions of a heading driven by hand are usually 8 treet 
high by 8 or 10 feet wide, but in solid rock where drilling 
machinery is introduced, it is advantageous to make the head- 
ing as wide as the tunnel at once. By drilling holes into the 
face at points about five feet each side of the centre, and con-~ 
verging on the centre line at a depth of about ten feet, a tri- 
angular mass of rock may be blown out, and the space thus 
gained facilitates the blasting of the adjacent rock on either 
side. An advance of about 10 feet in each day of 24 working 
hours may thus be made, using nitroglycerine in some form 
as the explosive agent. Owing, however, to unavoidable 
delays from various causes, this rate of progress cannot 
always be maintained. At the Hoosac tunnel the greatest 
advance in one week was 50 feet; in one month 184 feet at 
one licading. At the Musconetcong tunnel a heading 8 X 22 
feet in syenitic gneiss was advanced at the average rate of 
137 feet per month for 6 months, the maximum being 144 feet 
—the enlargement of the tunnel to full size going on at the 
same time, a few hundred feet behind. At the St. Gothard 
tunnel the north heading 2.5 x 8 metres was advanced in 
mica gneiss, during the year 1875 at the average daily rate of 
2-71 metres, with a maximum of about 4 metres, but the en- 
largement was not made. The south heading advanced at 
the rate of 2 metres a day, timbering being at times necessary. 

In ordinary clay a heading may be driven at from 795 to 180 
ft. per month, according to circumstances, where timbering is 
putin. The enlargement, inciuding timbering and masonry, 
may be advanced at from 20 to 60 ft. per month. Small tun- 
nels for water conduits are driven through dry clay at the rate 
of 10 ft. per day, the masonry following at once without tim- 
bering. 

The compressed air uscd to drive the drilling machinery 
serves to supply ventilation also. When this is wanting or 
proves insufficient, exhaust fans are used. At Mont Cenis a 


a ee 


pape FIELD ENGINEERING. 


horizontal brattice or partition was built in the tunnel, dividing 
it so as to secure a circulation of air. When foul gases are en- 
countered, ventilation becomes a serious question, and in one 
instance an important work was abandoned for this cause. 

Cross sections of the heading, and also of the tunnel en 
largement, should be measured at intervals of about 20 fect, as 
soon as opened, to see that the sides, roof, and floor are taken 
out to the prescribed lines, at the same time that the latter are 
exceeded as little as possible. In solid rock, since some ma- 
terial outside of the true section will necessarily be thrown 
down, leaving an irregular outline, it 1s well to take two cross 
sections at the same point, one following the projections and 
the other the recesses of the rock, from which an average sec- 
tion may be estimated. A daily, or at least a weekly, record 
of operations should be kept in tabular form, and the progress 
indicated by a profile and cross sections drawn on a sufficiently 
large scale to show dctails. 

The drainage of a tunnel is best secured by a line of 
stoneware or cement pipe laid in a trench along cach side, and 
covered with ballast or other loose material. The entire floor 
is thus made available for the use of the trackmen. - When an 
invert is used, the drain is placed in the centre between tracks. 
If the amount of water is large, drain pipe may be laid behind 
the walls, and the back of the arch may be covered with as- 
phaltum, or coal tar, to prevent a constant dripping on the 
track. 


251. Retracing the Line.—<As the grading pro- 
gresses, in either excavation or embankment, the principal 
transit points are established on the road-bed from the points 
of reference, and the centre line is retraced, setting stakes at 
every 50 fect. Transit points on grade should be fixed upon 
stout, durable posts firmly set in the ground, and standing 
high enourh to be easily reached after the ballast is laid. 
To recover the old line, any discrepancies in measurement 
must be Icft between the transit points where they occur, 
and not carried forward. In retracing a curve, if the transit 
is placed at the forward point, allowing the chain to ad- 
vance toward it, slight differences in measurement will not 
affect the position of the curve. If any short or long sta- 


CONSTRUCTION. 2Re 


tions have been introduced on the location, their position on 
the line must not be changed in retracing. The chain may 
be adjusted so that its measures will agree with the recorded | 
distances between transit points. Offsets are made right and Hi 
left from the new stakes to see that the road-bed is of the full | 
width at all points. The levels are also carried over the i) 
grade, and any remaining cut or fill found necessary is marked Wh 
on the back of the stakes, due allowance being made for the i | 
probable settlement of embankments. MM 


252. As the work approaches completion the contractor 
voes over the line dressing it to grade and opening the side i 
ditches if this has not been previously done. | 

Drain-tile should be laid at the bottom of these ditches and 
lightly covered with earth, particularly if the cut be wet. 
These not only prevent the water from reé ching the ballast, 
but by keeping the foot of the slope comparatively dry pre- 
vent the earth from sliding down and filling up the cut. 
There is also a marked economy in their use, as the cost 1s 
trifling, and all further excavation of mud and water from 
the cut is generally obviated. Should any springs appear in 
the slope a branch line of smaller tile may be laid to meet it. 
If the slope is liable to be overflowed from the surface above, 
an open ditch should be dug a few feet beyond the slope 
stakes, leading the surface water to discharge elsewhere. 


: 253. The road-bed being prepared, ballast stakes are 
| driven at every half station, giving the width of the ballast at 
its base, while the tops of the stakes indicate the proper level 
of its upper surface, which is the under side of the tie. These 
stakes should be set so as to give the proper elevation to the 
outer rails on curves when the ballast is graded to them. The 
ballast should be about one foot deep before the ties are laid. { 
Broken stone or a mixture of coarse and fine gravel is the | 
: best material, affording elasticity and good drainage. The 

side slopes of the ballast are made 1 to 1; its width at the Hl 

wider side of the tie should be one foot greater than the 

length of the tie. 


} 
: 
: 
254. Track-laying.—After the ballast has been laid 
and graded, the centre line is retraced upon it; short stakes 


224 FIELD ENGINEERING. 


are used, each of which is centred. On long tangents, one 
stake in every 200 fect is sufficient, on ordinary curves one in 
every 50 feet, and on very sharp curves one in every 25 feet 
The ties are then spaced evenly according to the number 
prescribed per mile, or per rail length; but a tie should not be 
allowed to cover a transit point. ‘Ties for the standard gauge 
are 8 or 9feet long; they should be sawed off square at the ends 
and in uniform lengths for appearance sake when laid. 
Specifications usually call for ties having a thickness of 6 
inches and a width of from 7 to 10 inches. The ends of the 
ties are aligned on one side of the road, though if cut into 
uniform lengths both ends will be equally well aligned. The 
rails are then laid on, and spiked to gauge. The first spikes 
are driven in the ties near’ a centre stake, the centre mark of 
the gauge bar being kept over the centre on the stake. Upon 
curves the rails must be sprung to the proper arc before they 
are laid (§199). Ad’ the ties required in a given distance 
should be laid before the rails are brought upon them. The 
practice of laying only joint and middle ties at first subjects 
the rails to the danger of bending from passing loads. 

Owing to the expansion of the rails by heat, a space 
must be left at the rail-joints. The highest temperature of a 
rail in the summer sun is about 180° Fah. The expansion of 
iron or steel per 100° is .0007 per foot; or for a 30-foot rail 
.021 foot or .262 inch, Therefore when 30-foot rails are laid 
at a temperature near the freezing point, or 100° below the 
maximum, the space allowed must be at least «a quarter of an 
inch. At 80° Fah. or 50° below the maximum, it need be only 
half as much. The space required is also proportional to the 
length of rail used. The exact space should be given, as less 
would result in the rails being forced up by expansion, while 
more than necessary space gives a rough road; and hastens 
the destruction of the rail. 

Wherever siding’s ‘are required, the necessary frogs and 
iong switch-ties should be provided in advance, so that they 
may be put in place at the time of laying the main track. For 
every road crossing at grade, heavy oak plank should be pro- 
vided, and laid upon the ties as soon as the rails are spiked, 
so that the highway travel may not be impeded. 


CALCULATION OF EARTHWORK. 


CHAPTER X. 
JALCULATION OF HARTHWORK. 


254. The first step toward finding the cubical content of 
an excavation is to divide it into a number of prismoids by 
several cross sections. 

A prismoid is a solid having plane parallel bases or ends, 
and bounded on the sides either by planes, or by such surfaces 
as may be generated by a right line moving continuously along 
the edges of the bases as directrices. 

The positions of the cross sections must be so selected 
that the solid included between any two consecutive sections 
may be a prismoid as nearly as possible. Upon a tangent the 
road-bed and side-slopes are planes, so that the prismoidal 
character of a given solid depends upon the shape of the natu- 
ral surface. When tlie natural surface is a plane, the sections 
are taken only at the regular stations, 100 feet apart; when it 
is curved, warped, irregular, or broken, the sections must be 
more numerous, so that the surface limited by any two shall 
be composed substantially of right-lined elements extending 
from one section to the other. 

If two end sections of a prismoid are somewhat similar, we 
infer that the corresponding points are connected by right- 
lined elements, forming in cach case the axis of a ridge or of a 
hollow. If one section has less breaks than the next, some of 
these ridges or hollows must vanish; and in order that the 
solid may be a prismoid, they must vanish in the section of 
least breaks; therefore a cross section must be taken on the 
ground through the point where each ridge or hollow vanishes, 
and the distance of that point from the centre line noted, so 
that it may be coupled with the proper point in the next section 
for exact.calculation of content. 

When ridges or hollows run diagonally across the line of 
road, cross sections must be taken where they are intersected 
not only by the centre line but also by the side slopes; that is, 
sections must be taken so that a side stake may stand on top of 


226 FIELD ENGINEERING. 


each ridge and at bottom of each hollow. In case the centre 
line intersects at right angles a retaining wall or other vertical 
surface, two cross sections are required at the same point, one 
at top and the other at base of wall, in order to furnish the 
data necessary to calculate the content each way from the ver- 
tical surface. (See Art. 235.) 

Every thorough cut terminates in either side-hill cutting, a 
pyramid, or a wedge; the latter happens only when the con- 
tour of the natural surface is at right angles to the line of road. 
Sections should always be taken through the points where the 
edges of the road-bed meet the surface, as these are the points 
of separation between thorough and side hill-work. Such sec- 
tions also serve to define terminal pyramids when they occur 
as is illustrated by Fig. 97. In side-hill work the foregoing 


Fie. 97%, 


rules apply as well, but sections will generally be more numer- 
ous than in thorough cuts. The same rules apply also to em- 
bankment, but as grading is preferably paid for in excavation; 
the same precision in determining the quantities in embank- 
ment is not usually necessary. 


CALCULATION OF EARTHWORK. 


255. Formule for Sectional Areas. 


Let 6 = base of section or width of road-bed, 


vertical 


: horizontal 
as — slope ratio = . 


‘‘ q@ = depth at centre stake. i] 
‘« h, k = depths at side stakes. Mi) 


‘‘m, m= horizontal distances from centre to side stakes. 


| 

| 

| 

I! 

For ground level transversely, the section is a parallelogram, ] 

and the area is evidently Wi 

j 

t 

| 


A = bd-+ sd? (342) 
or directly from the field notes, 
A=t+b0+tm+n)d (348) i 


For ground of uniform transverse slope between slope stakes, HHT) 


U ll i 

a ee 1 

See ee Od AV EE OM OES A Ni 

| J Hi 

| Hi 
h: | 
i | 
eh reles OE SHO ARIE SeCnIN tebe oe ii 

N A G 8 i 


Fic. 98. Hh 


Fig. 98, the section consists of the parallelogram 4 20H and | 
the triangle HOD. Hence | le. 


A= }(AB+ HO)EN + 41EODH — EX) 
A=}(AB. EN+ LO. DH) | 


or 
A= 3[bh +k + 2sh) | 
also (844) 
A = [dK + hid + sk) | 
From which also 
A = tbh + mk ) 
and (345) 


ee ahh eee 


These formule are independent of the centre depth. They 
are convenient for calculating the area of a plotted section 


228 FIELD ENGINEERING. 
having an irregular surface after the surface line has been 
averaged by stretching a silk thread over it. The points 
where the thread intersects the slope lines determine the 
values of h, &, m, and n respectively. 

When the ground has uniform slopes transversely from the 
centre to the side stakes: Fig. 99: If in the diagram we draw 


Qa 


——— foe ee ee ee 


ro 


Fia. 99. 


EG and DG, the section will be divided into four triangles, 
two having the common base CG =d and respective heights 
GN =m and GH=7n, and two having the equal bases AG = 
GB = and the respective heights HV =h and DH=k. 
Hence we have for the area of section 


A =4d(m+ n)+ 40h + &) (3846) 


Otherwise, if the slope lines are produced to meet below 


B. l 
1G = —¥ The area of CHPD is 


8 as 


grade at P, then GP= 


10P xX N= 4 («a -+ °) (m+ n). The area of ABP is AG X 
aS . 


b? 5 ‘ 
GP—=— Hence we have for the area of the section 


° 


9 


b\ b ~ 
Ae y(a+ =) (m+n) — tig (847) 


Both these formule are convenient, and as the values of the 
several letters can be substituted directly from the field notes, 
it ig unnecessary to plot such sections. 

When the surface of the ground ts irregular, verticals are con- 
ceived to be drawn to the grade line through the slope stakes, 


CALCULATION OF EARTHWORK. 229 


and through each break in the surface line, giving a number of 
trapezoids, the areas of which are severally calculated, and 
irom their sumis subtracted the area of the two triangles HVA 
and DiZBb. The remainder is the area of section required. 
This calculation may be made directly from the data furnished 
by the field notes without plotting; but if the ground has a 
number of small breaks, it is generally better to plot the sec: 
tions and stretch an averaging line over them, finding the areas 
by eq. (845). Or two averaging lines may be employed extend- 
ing from the centre stake, cach way, when the area may be cal- 
culated by eq. (846) or (847). 


256. Prismoidal Formule for Solid Contents. 
—The content of a prismoid may be exactly calculated by 
means of the Prismoidal Formula, which is 


Fie aia l ) AR ' 
8= gy At et 41 (348) 


S = cubic yards, / = length in feet, A, A’ = the areas at the 
two parallel ends, and MZ = the area of a section midway be- 
tween the ends. This area is not a mean of the other two, but 
the linear dimensions of the mid-section are means of the cor- 
responding dimensions severally of the end sections; from 
which therefore the area of the mid-section may be computed. 

The labor of calculating the middle area may be avoided in 
many instances by substituting in the prismoidal formula, eq. 
(848), for A, A’, and WM, their values as given in eq. (842) for 
ground level transverscly ; 


if 2 9 i he 
A=bd-+sd? A'=bd'+sd® M= pt zu d 4 ofl + ag 5.8 


i 
bt ED (SONY 9 Me Og u agit ! 
8 =9 27 [2sd ?+ (85 + 2sd')\d + (86 + 28sd')d'] (849) 


in which S is expressed in terms of the end dimensions, 


257. Tabies of cubic vards may be constructed upon this 
formula which are very convenient in practice. ° The constant 
values in any one table are / which 1s taken at 100, and 0 and s 
which are given values corresponding to the road-bed and slope 
ratio. The variables are d and d’. The columns im the table 


FIELD ENGINEERING. 


will be headed by the successive values of d', while each hori: 
zontal line will be headed by a value of d. For any one 
column therefore d' is constant, and the only variable is d. 
Assuming any value for @’, the values of S in that column may 
be computed, letting d take a series of values differing by unity 
from zero upwards, and the corresponding values of S will be 
placed in the column d’ opposite the several values of d. 

But instead of solving the eq. (849) for each value of S re- 
quired, the process of filling the table may be much abbre- 
viated by observing that since the equation is of the second 
degree with respect to the variable d, the second difference of 
the values of S will be a constant and equal to twice the co- 

Ast 
6 X 27 
of first differences of S in the column d’ (é.e. between d = 6 
and d =1) is expressed by the swm of the coefficients of @ 
and d; or 


efficient of @’, or 6" = Also the first term in the series 


60 = 


U 9 
b+ 281-4 d' 
The first value of S in any column d’ is found by solving 
eq. (349) after making d= 0; or, 
Deore go8 bis [ (8b + 28d "\d"] 
6 X 27 
Starting with these values we may fill any column d’ simply 
by successive additions. The values of d’ for the several 
columns should also differ by unity. The final value of S in 
each column should be calculated by formula as a check; or 
since all the final quantities in the same line d of the table 
form a series of which the second difference is 6", if on taking 


their differences this result is obtained, the quantities are 
proved to be correct. 


Example.—Given a base of 18 feet and slopes 14 to 1, to fill 
the column of d' =6 in a table of cubic yards for level cross 
sections. Here [=100, 5=18, s=3, d'=6. Hence d’= 
8.70874, by’ = 46.2962, and S, = 266.6666-+. It is not 
necessary to go beyond the fourth decimal place, since that 
figure will always be the same as the first decimal (a result 


CALCULATION OF EARTHWORK. 20% 


due to dividing by 27), and may be corrected by it after every 
addition. The process is as follows: 


d S 6' 6” 
0 266.6666 
1 312.9629 46.2962 3.7087 
2 362.9629 ed ake 3.7087 
3 416.6666 pesdue 3.7037 
, NYA 57.4074 9 RENE 
Biers 61.1111 aac 
6 600.0000 64.8148 3.7037 
% 668.5185 68.5185 3.7037 
1 668.5185 72, 9999 3. 
8 740.7407 eee 3.7037 
etc. ete. etc. 


In copying into the table, the quantities are taken to the 
nearest unit only, and the decimals are otherwise neglected. 

The completed table furnishes values of 8 corresponding to 
any values of d and d@' in even feet. The correction for the 
decimal parts of the depths, when there are such, is made by 
adding to S (found opposite the even feet) the product of the 
half sum of the decimals by the difference between S as 
found and the next value of S diagonally below to the right. 
These differences may for convenience he inserted originally 
in the table under each quantity in small figures. 

If the length of the solid differs from 100 feet, multiply the 
corrected quantity by the length and divide by 100, since S 
varies directly as 7. Such tables are published in separate 
sheets for a variety of bases and slopes, so that usually one 
may be purchased to suit the case in hand. 


258. These tables may be used to find quantities when 
the ground is not level transversely, by finding, first, the area 
of the actual sections, and second, the depths of level sec- 
tions having equal areas, and then using the depths so found 
as the values of d and d' in the table of quantities. The 
depths of equivalent level sections are called equiya. 
lent depths. They may be calculated by the formula 


b ‘A: pt? 
SS tebe / emit Ps; (350) 
which is derived directly from eq. (842). The more convenient 


method, however, is to construct a table on eq. (842), giving to 
d a series of values varying by one tenth of a foot from zero 


FIELD ENGINEERING. 


upward. ‘The values of 8 and s in this table must agree with 
those of the road and of the table of cubic yards. 


259. When the transverse slope is uniform between slope 
stakes the equivalent depth may be expressed in 
terms of the centre depth and slope of surface 
without reference to the area, Fig. 100. 


D 


Fie. 100. 
Let HABD be the given section. 
‘* RABT be the equivalent level section. 
Produce the side slopes to mect at P, and let c = CP and 
Co OrP3 
Through C draw the horizontal line QZ, and at ZL erect the 
perpendicular LM = z, and draw LN parallel to PL. 
The area RPT = area HPD; and QHC= LNC; hence area 
EPLN = QPL = EPD — NLD. 
Since VLD is similar to HPD, we have 
HPD NALD, 33 Cae 
or HPD — NID: HEPD:: 2-2: ¢ 
Since YPL and kPT'are similar, 


OPE s BPL tO Hee 


cA 
oF — 22:02 3: cs ¢,2 or 4? = ——— 
oe — 2 
A CL es c? 8? 
Let s' = slope ratio of surface = —— =—. Then2 = —- 
ML 2 8 


which substituted gives 


eet raeeen (351) 


eee 
V3'2 — 3? 


CALCULATION OF EARTHWORK. 


If d, = the equivalent depth C,G, then 
di =d+(Q— ¢) 


A table may be prepared giving (c: — ¢) for various inclina- 
tions of surface with given base and side slopes. It is then 
only necessary to add this correction to the centre depth to 
obtain the equivalent depth. Such tables of correction usually 
accompany the published tables of cubic yards. This method 
of obtaining quantities is particularly applicable to preliminary 
estimates, where the ground has not been cross-sectioned, and 
only the centre depth and transverse inclination is known. 


260. The use of the earthwork tables described gives 
correct results ;— 

1st. When the surface of the prismoid is a plane, however 
much inclined; provided it does not intersect the road-bed 
within the limits of the prismoid. 

2d. When with regular, or three-level end sections, generally 
similar to each other, the surface is regularly warped from one 
end to the other; provided that the side lines and centre line 
of the surface are straight, and that no two of them are in- 
clined to grade in opposite directions. 

3d. When the ridges or hollows of an undulating surface 
are parallel to the line of the road. 

Ath. When a surface of numerous irregularities may be 
averaged by planes or warped surfaces so as to comply with 
one of the preceding conditions. 

But the method fails on undulating ground when the 
ridges or hollows run obliquely to the line of road, even 
though the sections may appear quite regular. 

In general, the method of equivalent depths holds good 
when the mid-section of the equivalent level end sections 
equals in area the actual mid-section or the prismoid; other- 
wise it fails. 


261. The content of a prismoid may be approximately ob- 
tained by the method of mean areas, the formula for 
which is 


y 
S= - A + A! (352) 


2x 27 


Although approximate, this method is much employed on 


ro4 FIELD ENGINEERING. 


account of its convenience. It is approved by statute to be 
used upon the public works of the State of New York. 

If the values of A and A' derived from eq. (842) be substi- 
tuted in eq. (852), and then eq. (849) be subtracted from it there 
remains ' 

3 
exer 74) 
Hi which is the correction by which S obtained by eq. (852) must 
i be diminished to make it equal to S obtained by eq. (349) 
when the ground is level transversely. 
Again, for three-level sections, if the values of A, A’, and M 
derived from eq. (847) be substituted in both eq. (848) and 
(352), and one subtracted from the other, there remains 


a 1 ; ae , na 
Av 1237 [(d — d') (m+n —(m'+-n')) ] 


which is the correction by which S ubtained by eq. (352) must 
be diminished to make it equal to S obtained by eq. (849). 
Hence we may write at once, for three level-sections, the cor- 
rect formula: 


Mi 4 bs [m+n — on'-+n 1 | (853) 


& 


s l 


This formula gives results identical with eq. (349), is applica- 
ble to the same cases, and gives correct results or fails to do 
so according to the conditions stated in the previous section. 


262. When the conditions of the surface are such that eq. 
(349) or eq. (853) will not give correct results, the area of 
the mid-section may be derived from its calculated linear 
dimensions as stated in § 256. The contents of the prismoid 
are then given by eq. (848), 


Example 1. (Fig. 101.)—Base 20. Slopes 14 : 1. 


22 0 7.5 
ae el a A oe ee -°, Aas 443 gn ft 
i} 84 0 16 


| A a gee: + ig ". A = 200 sq. ft. 


CALCULATION OF EARTHWORK. 


If 7 = 100, eq. (848) 


Miccc-sty 161s pe Sele 
S = Gray (448 + 967 + 200) = 1001 c. yds. 


Had this been calculated by eq. (849) or eq. (853) or by the 


Fie, 101. 


tables, the result would be 1167 c. yds., showing an error of 
166 cubic yards in excess. 


Heample 2. (Fig. 102.)—Base 20. Slopes 14: 1. 


22 0 19 
Af ae 7 + re 234 sq, ft. 
et pe le 
13 0 16 
ij tt - v' + 7d 88 sq. ft. 
17.5 11 0 8 17.5 sad 
M+ ef : a5 _ + ana can 164.5 sq. ft. 


If 7 = 100, eq. (848) 


y 100 5 > oc (‘Viva ad 
‘Pies ora [88 +- 658 + 234] = 605 c. yds. 


Had this been calculated by eq. (849) or eq. (353) or by the 


296 FIELD ENGINEERING. 


tables, the result would be 584 c. yds., showing an error of 
21 cubic yards in deficit. 


263. At the termination of acut or fill we have usually 
either a wedge or a pyramid. To a wedge the pre- 
ceding formule and tables based on them apply by makiny 


Fie. 102, 


one end depth equal zero. In the case of a pyramid, the 
content is equal to the area of the section forming the base 
multiplied by one third the length of the solid, and divided by 
24: OF | 
LA 


Oe KOT 


(354) 

264. Side-hill Work.—When the natural surface has 
a regular transverse slope and intersects the road-bed, the 
cross section is reduced toa triangle. If w = the intercepted 
portion of the road-bed, and & = the side height, then A = 
tok. Similarly A’ = 40% and 4M = 4w+w’) (A+ hb), 
which substituted in eq. (848) give 


t l 

S = —— ~~ (Qu + wk 2w' + w) k' (355 
which is convenient for direct calculation from the field notes. 
It is not adapted to the construction of tables, since it contains 
four independent variables, 


CALCULATION OF EARTHWORK. 

If the slope of the natural surface is given, let s' be the sur- 
face slope ratio at one section, and s” that at the other, and s 
the ratio of the side slope. Then w = k(s' — s) and w' = 
k'(s" — s), which substituted in eq. (855) give 


Y U 4t im) " ae s' a " ¥] 
S = 53087 (s'— s)k? + EUs eras kk + (8 ead 8) k? 
If the surface is a plane, then s" = s', and we have for this 
case 
fa ee he tk] (356) 
PG. 627 


which is a formula of quite limited application; yet it is the 
one on which tables and diagrams are usually constructed. 
Consequently the latter will not give correct results, except 
when the surface is a plane. 


265. When the natural surface is broken the 
sections may be plotted, and the values of «# and & taken from 
the points where an averaging line intersects the grade 
and side slope respectively. Finding values for w' and x’ in 
the same way, the content may then be obtained by eq. (855) 
as before. The averaging line should not only cut off the 
same area as the original section, but should also have in cach 
case a slope agreeing as nearly as possible with the general 
slope of the natural surface. The slope is determined simply 
by inspection of the diagram, but the area may be bad pre- 
cisely, for, taking w from the averaging line, and knowing A, 
we may calculate k by the formula k = = or k may be 
taken from the plot and 2 calculated. 

Otherwise, the actual mid-section may be calculated and 
the cubic contents determined by the method illustrated in 
§ 262. 


266. To express side-hill areas and cubic yards 
in terms of the centre depth, d, and transverse 
slope-ratio s’. Fig. 103. 


5? 


Whend=9, A= 0k = =p 


FIELD ENGINEERING. 


For any depth d, add to this area 


; Reet : b sd 
sa (e+ 5) = 4a ee 5 ta@ a) 
and there results, 


_@o+sa" 
=e pomens it 


_ Wab+e'a? 
—~ 2X 278" —8) J 


(857) 


Wh Observe that d may be plus or minus, and that its limits are 
WE b 

a Tables of cubic yards may be constructed on this formula, 
i HH making d and s’ the variables, which would be extremely con- 


Fie. 108. 


venient for making up estimates upon preliminary lines on 
which the profile of centre line and angle of transverse slope 
Hil only are known. Since s’ is the cotangent of the slope angle 
| the columns of the table may be headed by the angles in a 
i series of degrees, while the corresponding values of s' are 
Hi used in the formula. The values of d@ should vary by tenths 
i] of afoot. The results obtained by eq. (356) and eq. (857) will 
be identical for the same sections. 


267. Several different systems of diagrams have been 
devised and published for determining quantities in earthwork 
by a sort of graphical method. These diagrams, which are 
substitutes for tables are preferred by some engineers. They 


CALCULATION OF EARTHWORK. 239 


are based on the same principles, and are constructed on modi- 
fications of the same formule. 


268. Correction of Earthwork for Curvature. 
—The preceding calculations are based on the assumption that 
the centre line is straight, with cross sections at right angles to 
it. When an excavation is on a curve, the cross sections, be- Hh 
ing in radial planes, are inclined to each other, so that the con- Hl 
| dition of a prismoid is not exactly fulfilled. But by the proper- | 

ty of Guldinus, if any plane area is made to revolve about an 
axis in the same plane, the volume of a solid generated by the 
area is equal to that of a prism having a base equal to the given | 
area, and a height equal to the length of path described by the 
: centre of gravity of the area. The path, being the arc of a cir- 
cle, is proportional to the radius drawn to the centre of gravi- Wma 
ty. If therefore a cross section is symmetrical with respect to t 
the centre line, the path of the centre of gravity is equal to the Hi | 
measured length of the centre line, and no correction for cur- 1 
vature is required. 

But when the ground is inclined transversely, the centre of 
gravity is one side of the centre line, and its path, if we con- i 
: ceive it to sweep around the curve, from one end of a prismoid 
to the other, is longer or shorter than the distance measured on 
the centre line, according as the centre of gravity is outside or 
inside of the centre line curve. 


Let C = correction in cubic yards due to curvature. 
‘« S = cubic yards as obtained by prismoidal formula. ii 
‘*. R= radius of centre line. 
“* é-= eccentricity of centre of gravity of section. ai 

= horizontal distance from centre line to centre of i] 
gravity. HI} 


We then have the proportion, | 


S+ Os: Sh te6:R 


As the sections of a solid are seldom similar and equal, we 
shall usually have a diffcrent value of e for every section, from 


240 FIELD ENGINEERING. 


which, however, a mean average value may be deduced, and 
used in the above formula. But it will be more convenient to 
correct the areas themselves for eccentricity before finding S, 
which will then require no correction. For the same result 


will ensue whether we multiply S by 7 or multiply one of 


the component factors of S by the same ratio. 
If then ¢ = correction of area in square fect due to eccentri- 
city, we have at once 


Ae 
R 


and the corrected area equals A + ¢ according as the cut is 
deeper on the outside or inside of the curve. Each area used 
in determining the solid contents should, on a curve, be first 
corrected in this manner, 


To find the value of e for any three-level section, Fig. 104. 


& 


+ 


ss Ta ee 0 ws ew anal ee ee 


% DERISION IAL yp 


Fie. 104. 


i 
| 
i 
I 
' 
I 
\ 
\ 
j 
1 
t 
| 
| 
I 
u 


Find the areas either side of the centre line separately, call- 
ing them Hand X, and take their sum and difference. Using 
the same notation as in § 255, H = md + ibh, K = ind kik 
bk, and + K = A, 


K —H = 4d (n — m) + 30 (k —h) 
In the figure draw CZ’ equal to O#, and the triangle CH’D 


will represent the area (K — #7). Bisect the side E'D, and 
draw a line from (@ to the middle point. Then the centre of 


CALCULATION OF EARTHWORK. 2AI 


gravity of the triangle will be on this line at two thirds its 
length from @, and the horizontal distance of the centre of 
gravity from C is ? xX m+n) = 3(m-+n). 'The centre of 
gravity of the remainder of the section is on the centre line 
CG, so that the value of ¢ is found from the proportion 


€é:¢t(n+m): K-—H:A 


_n+t+m _,, 
ein 7 (K — ff) 


| | Ae nm +m 
: Hence c= Peas a [$d (n —m) +4) (k—h)] (858) 


Sections which are more irregular may be plotted and 
reduced by averaging lines to three-level sections, in order that 
the formula may be applied. If the ground is so irregular as 
to require the computation of the middle section, the correc- 
tion ¢ should be found and applied to this area (M) also 
before introducing it into the prismoidal formula. As the 
correction for curvature is always relatively small, it is usually 
ignored in practice for thorough cuts, except where deep cut- 
tings with steep transverse slope occur on sharp curves. 

The correction is of more importance relatively in 
side-hill work as the centre of gravity of the section is 
more remote from the centre line. Let the section be reduced 


A ce 


Fre. 105. 


to a triangle by an averaging line (Fig. 105), and w be the base 
of the triangle formed by the averaging line. The centre of 
gravity is at one third the horizontal distance from the middle 
point of 2 to the side stake D, while the distance of this 
middle point from the centre stake C is evidently 40 — fe. 


FIELD ENGINEERING. 


Hence ¢ = 3b — lwo + 3[n — GO — Ww) 
e=db-+n—w) 

_ Ae b4+n—w, wk f 
and ¢= R = 3R OK 5 (859) 


The correction ¢ will be plus or minus as before explained. 
This formula applies to all side-hill triangular sections, 
whether there be cut or fill at the centre stake 


Example 1.—Thorough cut; base 20; slopes 14: 1. 


1 = 100; 8° curve, left; R = 716.78 


16 12 58 
Notes. A.+ mn + 6 + 39 


: Powe 8 _ 40 
Gran ee ba GRIST 
Then K = +X 58 X 12+4 X 20 X 82 = 508 
H=4x16X%12+4X20X 4=116... A= 624 


K—-H= 392 
ye g 
Kq. (858) ¢ = Ra! awe 5 chee 13.49 


(A+ c) 687.49 
K'=:i4x40x8tt x 20 X 20 = 260 
Ho = F318 X8 +X 20X 2= 82. aoe 
Ki He 198 


2 AR PaO e | 
© = 330 716.78 = 4:86 


(A’ +c’) = 826.87 
From which we obtain S = 1758 cub. yds.—Ans. 


Without correction we have 1726 ‘‘ A 


32 66 66 


Showing a difference of 


Had the curve been to the right with same notes, ¢ wouid 
have been minus, and S would = 1694. 


CALCULATION OF EARTHWORK. 245 


Example 2.—Side-hill cut; base 20; slopes 14 :1 
= 60; 10° curve, right; R = 573.69 


: 6 iPaper fh | 
tg 0 +28 T 39 i| 
0 2 ist BY | 
0.80.01 is i) 

A=} x16 x20 = 160 |i 

2 as i) 
MT OR oa i ey 7m 3.58 | 


3 X 573.69 


(A —c) = 156.42 


A'=4x8x18= 72 i 
pein QO OT — Bieks i 
== ne, Oa 9.05 I 
© = 3 x 573.69 Ye | 


Hence S = 248 cub. yds. if 
Without correction Swould = 255 “ « 


Difference © joes y 


269. Haul.—The cost of removing excavated material, I 
when the distance does not exceed a certain specificd limit, is | 
included in the price per cubic yard of the material as meas- 
ured in the cutting. But when the material must be carried 
beyond this limit, the extra distance is paid for at a stipulated 
price per cubic yard, per 100 feet. The extra distance is known il 
by the name of haul, and is to be computed by the engineer 
with respect to so much of the material as is affected by it. iy 

The contractor is entitled to the benefit of all short hauls 
(less than the specified limit), and material so moved should not 
be averaged against that which is carried beyond the limit. 
Therefore, in all cuts, the material of which is all deposited 
within the limiting distance, no calculation of haul is to be 
made. 

On the other hand, the company is entitled, in cases of long 
haul, to free transportation for that portion of the cutting, no 
one yard of which is carried beyond the specified limit. . There- Hot 
fore, this portion is first to be determined in respect to its ex- 
tent; and the number of cubic yards contained in it is to be de- 


R4A+ FIELD ENGINEERING. 


ducted from the total content of the cutting, before estimating 
the haul upon the remainder. Find on the profile of the line 
two points, one in excavation, and the other in embankment, 
such, that while the distance between them equals the specified 
limit, the included quantities of excavation and embankment 
shall just balance. These points are easily found by trial, with 
the aid of the cross sections and calculated quantities, and be- 
come the starting points from which the haul of the remainder 
of the material is to be estimated. 


Fic. 106, 


Fig. 106 represents a cut and fill in profile. The distance 
AB isthe limit of free haul. The materials taken from AO 
just make the fill OB and without charge for haul; but the haul 
of every cubic yard taken from AQ, and carried to the fill BD, 
is subject to charge for the distance it is carried, less AB. It 
would be impossible to find the distance that each separate yard 
is carried, but we know from mechanics that the average dis- 
tance for the entire number of yards is the distance between 
the centres of gravity of the cut AC, and of the fill BD which 
is made from it. If, therefore, X and Y represent the centres 
of gravity, the actual average haul is the sum of the distances 
(AX-+BY), and this (expressed in stations) multiplied by the 
number of cubic yards in the cut AC, gives the product to 
which the price for haul applies. 

But the product of AX by the number of cubic yards in AC 
is equal to the sum of the products obtained by multiplying the 
contents of each prismoid in AC by the distance of its own 
centre of gravity from A. The distance of the centre of gravity 
of a prismoid from its mid-section is expressed by the formula 


eA A 
v= 987 8 CY) 
. l 4 a j . 
If we replace S by its approximate value, ee which 
will produce no important error in this case, we have 
ll A-—A 
x= (361) 


6 Ae 


CALCULATION OF EARTHWORK. 245 


in which A should always represent the more remote end area 
from the starting point A, fig. 106. Hence, may be + or —, 
and it must be applied, with its proper sign, to the distance of 
the mid-section from the starting point A, before multiplying 
by the contents S. Each partial product is thus obtained. 

By a similar process with respect to the prismoids composing 
the mass BD, and using the point /as the starting point, we 
obtain finally a sum of the products representing this portion 
of the haul. 

If a cut is divided, and parts are carried in opposite direc- 
tions, the calculation of each part terminates at the dividing 
line. Ifa portion of the material in AC is wasted, it must be 
deducted, and the haul calculated only on the remainder. 

The specified limit is sometimes made as low as 100 feet, 
sometimes as high as 1000 feet. A limit of about 300 feet, how- 
ever is usually most convenient, as it includes the wheelbarrow 
work, and a large part of the carting, while it protects the con- 
tractor on such long hauls as may occur. 


270. The Final Estimate is a complete statement in 
detail, of the amount of work done and materials provided, in 
the construction of the road, and is the basis of final settlement 
between the company and contractor. Its preparation should 
be begun as soon as possible after the work is in progress, and 
should be continued, as fast as the necessary data are accumu- 
lated, while the circumstances are still fresh in mind, and when 
any omissions in the field notes may be readily supplied. The 
content of each prismoid, the classification of its material, and 
the length of haul to which it is subject, should be matters of 
special record in a book provided for that purpose. These re- 
sults having been carefully computed by exact methods form 
a standard of comparison for those approximate results which 
must be had from time to time during the progress of the work, 
and furnish a limit to the amounts of the monthly estimates. 
The same remark applies to all other items of labor and mate- 
rial. The notes and record of the final estimate should be par- 
ticularly full and exact in respect to all such items as will be 
inaccessible to measurement at the completion of the work, 
such as foundation pits, foundation courses of masonry, cul- 
verts, and works under water. 


246 FIELD ENGINEERING. 


271. Monthly Estimates.—On or before the last day 
of every month during the progress of construction, measure- 
ments are taken to determine the total amount of work done 
and material provided up to that date. The estimates based 
on these measurements are called Monthly Estimates. It is fre- 
quently necessary to take measurements for both monthly and 
final estimates at other times than the end of the month, as in 
the case of foundations which are not long accessible. With 
respect to each piece of work satisfactorily completed, the 
monthly estimate should be exact, and identical in amount 
with the final estimate. With respect, however, to items of 
work in progress at the time of measurement, the monthly 
estimate is only approximate, yet should be as precise as the 
nature of the case will allow; and the quantities stated should 
not be in excess of fair proportion of the total quantities given 
on the final estimate for the same piece of work. 

A special field book is devoted to monthly estimate 
notes. Each page should be dated with the day on which the 
notes upon it were taken. The notes consist of measurements 
of all sorts, principally of cross sections partially excavated. 
These sections should be at the same points on the line as the 
original sections, so that comparisons may be made. Where- 
ever the excavation is finished to grade, it is only necessary to 
write ‘‘completed” opposite such stations, and the quantities 
may be taken from the final estimate or computed from the 
original notes. It is frequently necessary to retrace portions 
of the centre line in taking estimate notes, so that all the field 
instruments are required, but a party of three or four men is 
usually sufficient. 

If the contractor has provided materials, such as stone, lum- 
ber, etc., which are not as yet put into any structure when the 
estimate is taken, these should be measured and entered under 
the head of temporary allowance, an arbitrary price be- 
ing used somewhat below the actual value of the material as 
delivered. Such allowances should never be copied from one 
month’s estimate to the next, but made anew on such material 
as may be found that seems to require it. But all completed 
items of contract work, and of extra work when ordered by 
the engineer, are necessarily copied from one monthly esti- 
mate to the next during the continuance of the contract. 

A blank form is used by the resident engineer in report 


TOPOGRAPHICAL SKETCHING. 24AG 


ing monthly estimates, on which a column is provided for each 
class of material and work required by the contract, while the 
several lines, headed by the numbers of the proper stations, are 
devoted to the different cuttings, structures, etc., in consecu- 
tive order as they occur on the line of road. The estimates are 
made out and reported separately for the several sections into 
which the line of road is divided for letting. 

These reports are reviewed by the division engineer, and 
the footings copied upon another blank, which is the monthly 
estimate proper; the prices are attached to the items, and the 
amounts extended and summed up. This sum indicates ap- 
proximately the total amount earned by the contractor up to 
date, fron. which is deducted a certain percentage (usually 15 
per cent.), which is retained by the company until the comple- 
tion of the contract. From the remainder is deducted the 
amount of previous payments, which leaves the amount due 
the contractor on the present estimate. A blank form of re- 
ceipt is appended, to be signed by the contractor. 


CHAPTER XI. 


TOPCGRAPHICAL SKETCHING. 


272. Topographical sketches taken on preliminary surveys 
are usually of great value in projecting a line for location; 
they should be made therefore as accurate and complete as 
possible. In too many instances sketches are presented having 
a picturesque appearance, but conveying little information, if 
not tending to mislead the map-maker. The aim of the topog- 
rapher should be to record the topographical features either 
side of the line with as much precision as those directly upon 
the line, without taking actual measurements, except in rare 
instances. The eye and the judgment must be usually depended 
on for distances and dimensions. The sketch of a tract ex- 
tending to 400 feet each side of the line ought to be accurate 
enough to warrant its being copied literally upon the map. If 
a much wider range is required it may be advisable to use the 
plane-table; but an approximation to plane-table methods may 
be employed in ordinary sketching. 


FISLD ENGINEERING. 


273. As artificial features are the most readily de- 
fined and located these should first receive attention in making 
a sketch. When recorded they form a skeleton upon which 
the natural features can be drawn with more precision than if 
the order were reversed. ‘The point where each fence crosses 
the line and the angle between the two may be sketched exact- 
ly. The distance along the feuce to any object may be esti- 
mated, and checked (in case of an oblique angle) by observing 
where a line from the object perpendicular to the centre line 
would intersect the latter. The book may be rested on any 
support, the centre-line of the page coinciding with the line of 
survey, and the direction of objects defined by a small ruler 
laid on the page. This operation being repeated from another 
point gives intersections which locate the several objects on 
the sketch. If the bearings are taken they may be plotted on 
the page as well as recorded, giving the same results. The 
eye may be trained to estimate distances Correctly by observ- 
ing the appearance of objects along the measured line, the dis- 
tances to which are therefore known. 


274. After the artificial objects the more distinct natural 
features are to be sketched, as streams, shores, margins of 
swamps, forests, etc., ravines, ridges, and bluffs, taking care 
that all these outlines intersect the features of the sketch 
already delineated at the proper points. The correct repre- 
sentation of contours is the most difficult part of sketching, 
since these lines are quite imaginary, yet for railroad maps 
they are usually as important as any others. It is desirable to 
know not only the locality of a hill or slope, but also its shape, 
steepness, and height. This information is best given by con- 
tour lines. A contour 1s the intersection of the surface of the 
ground by an imaginary level surface. When the surface 1s 
real, like that of a lake, the intersection 1s called ashore. If 
the water should rise a certain -height a new shore would be 
defined, and rising double that height still another shore 
would result, each of which, on the subsidence of the water, 
would be a contour. <A practiced eye is able to follow on the 
ground the course of a contour with all its windings; but in 
sketching them due allowance must be made for the fore- 
shortening effect of distance. All contours on the same sketch 
should have the same vertical interval, so that by counting 


TOPOGRAPHICAL SKETCHING. 24% 


them the height of the hill may be known. The spaces on the 
sketch between contours vary as the cotangent of the slope 
angle, so that the width of the spaces indicates the degree of 
steepness. ‘The contours nearest the topographer should gene- 
rally be sketched first, although if there be a shore that is apt to 
be the best guide to the share of the slopes. Ifthe height of the 
hillis known and the upper contour located, the other contours 
can be spaced between with less difficulty, the proper number 
being ascertained by dividing the height by the assumed verti- 
cal interval. A special line of levels up an inclined ravine or 
sloping ridge to fix the contour points is often of the greatest 
service in obtaining correct results. Other random lines are 
sometimes run to locate the contours more definitely. These 
should be made to cross several contours rather than to trace a 
single one. Old preliminary lines which have proved useless 
in themselves often furnish by their profiles valuable informa- 
tion 1n respect to contours. 

The use of hatchings should be avoided in the sketch-book, 
except to represent precipitous banks, or slight terraces, which 
would not be sufficiently defined by the contour system. 
Hatchings frecly used consume too much time, and fail to give 
an accurate idea of either slope or height, while they obscure 
the page for the representation of other objects. 


275. The centre line on the page is straight, and for 
sketching purposes the surveyed line on the ground is assumed 
to be so also. = Slight deflections in the course of a preliminary 
line may be ignored 1n the sketch; but if a large angle occurs 
it is better to terminate the sketch with the course, and begin 
again, leaving a few blank lines between the two sketches. 
On a located line with curves, the sketch is continuous. The 
curved line in the field is represented by the straight line on 
the page, and the radial lines through the stations are repre- 
sented by the parallel lines ruled across the page. Al objects 
vive sketched at the proper offset distance by scale frem the 
centre line; but longiiudinally the sketch 1s necessarily dimin- 
ished outside of the curve, and magnified inside of the curve. 
Consequently topographical lines which are straight in fact ap- 
pear curved in the sketch, concave to the centre line if inside 
the curve, and convex if outside of it. Such features are cor- 
rectly sketched by means of offsets estimated or measured 


250 FIELD ENGINEERING. 


from each station of the curve on the radial lines. This kind 
of distortion creates no confusion if properly done, for in mak- 
ing the map, after drawing the curve and the radial lines, the 
same offsets will give the correct positions of the objects delin- 
eated. This method is preferable to drawing a curved line on 
the page to represent the centre line, as it is difficult to draw 
it correctly; it will cross the ruled lines obliquely, rendering 
them of no service for offsets or scale, and moreover is likely 
to run off the page altogether. 


CHAPTER XII. 


ADJUSTMENT OF INSTRUMENTS. 


Every adjustment consists of two processes: first the test, and 
second the correction. Inasmuch as the amount of correction 
is made by estimation, the test must always be repeated until 
no further lack of adjustment is observable. 


276. THE TRANSIT. 

The level tubes should be parallel to the 
vernier plate. 

Test : Place the tubes in position over the levelling screws, 
and turn the latter till the bubbles are centred; revolve the 
plate 180°. The bubbles should remain centred; if they have 
retreated— 

Correction : Bring them half way back to the centre by 
turning the adjusting screws which attach the tubes to the 
plate. 


The line of collimation should be perpendi- 
cular to the horizontal axis. 

West: Clamp the limb, and by the tangent screws bring 
the intersection of the cross-hairs to cover a well-defined point 
about on a level with the telescope; plunge the telescope to 
look in the opposite direction, and note any point about on a 
level with the telescope and about equidistant with the first 
point, which the intersection of the cross-hairs now happens to 
cover. Now unclamp the limb and turn through 180°, and 
repeat the above operation, using the same first point as before. 


ADJUSTMENT OF INSTRUMENTS. 201 


The tiird point obtained should be identical with the 
second, if not— 

Correction : Move the vertical cross-hair over one fourth 
of the apparent distance from the third to the second point, by 
turning the adjusting screws at the side of the telescope. 


The horizontal axis should be parallel to the 
vernier plate. 

Test; After completing the above adjustments level the 
limb, clamp it, and bring the intersection of the cross-hairs to 
cover some high point so that the telescope may be elevated to 
a large angle; depress the telescope and note some point on the 
ground now covered by the intersection of the cross-hairs. 
Now unclamp the limb, turn it through 180°, and repeat the 
above operation, using the same high point as before. The 
third point found should be identical with the second; if not— 

Correction: Raise the endof the axis opposite the second 
point (or lower the other end) by a small amount, by turning 
the adjusting screws in the standard. The amount of motion 
required is only determined by repeated trials until the test is 
satisfied. 


The intersection of the cross-hairs should 
appear in the centre of the field of view. 

Test ; Bring the cross-hairs into focus and direct the tele- 
scope toward the sky, or hold a sheet of blank paper in front 
ofit. If the intersection appear eccentric— 

Correction ; Turn the screws (by pairs) which support 
the end of the eyepiece until the desired result is obtained. 


If there be a level on the telescope it Should be 
parallel to the line of collimation. 

Drive two stakes equidistant from the instrument in exactly 
oposite directions, and having perfected the previous adjust- 
ments, level the plate carefully, Glamp the telescope in about 
a horizontal position, and observe a rod placed on each stake. 
Have the stakes driven by trial until the rod reads alike on 
both. The heads of the stakes are then on a level.  Re- 
move the instrument beyond one stake, and set it up in 
line with the two, level the plate, and elevate or depress the 
telescope to a position which will again give equal readings 
on the stakes. The line of collimation is now level— 


2D2 FIELD ENGINEERING. 


i) Test: While in this position the bubble of the attached 
level should stand centred; if not— 

Correction: Bring the bubble to the centre by turning 
the nuts at one end of the tube, while the cross-hair continues 
to give equal readings. | 

277. THE Y LEVEL. | 


The line of collimation should coincide with 
the axis of the telescope. 

Test: Clamp the spindle, and bring the intersection of the 
cross-hairs to cover a well-defined point by the tangent and 
levelling screws; revolve the telescope half over in the Ys, so 
Hil) that thelevel tube is on top. The intersection of the cross- 

HE hairs should still cover the point. If either hair has departed— 

Pi Correction: bring it half way back by means of the pair 
' ly oi adjusting screws at the extremities of the other hair. 
agit The attached level should be parallel to the 
axis of the telescope. 
i Test: Bring the telescope over one pair of levelling screws, 
i, clamp the spindle, open the clips, and bring the bubble to the 
| centre. Then gently remove the telescope from the Ys, and 
replace itend forend. Ifthe Ys have not been disturbed, the 
bubble should return to the centre. If it does not— 

Correction : bring the bubble half way back by turning 
the nuts at one end of the tube. 


But as now the level tube and telescope may only lie in 
parallel planes, and yet not be parallel to each other— 

Test: bring the bubble to the centre as before, and turn 
the telescope on its axis so as to bring the level tube out to one 
‘il side. The bubble should remain centred. If it has departed— 
it Correction : bring it back to the centre by the adjusting 
ji screws at one end. 


1 The axis of the telescope should be at right 
i angles to the spindle. * 

Test: Having completed the above adjustments (and not 
before), fasten down the clips, unclamp the spindle, and bring 
the bubble to the centre over each pair of levelling screws in 
succession, then swing the telescope end for end on the spin- 
dle. The bubble should settle at the centre. If it donot— | 

Correction: bring it half way back by the large nuts at 
one end of the bar. 


EXPLANATION OF TABLES. ROO 


278. THE THEODOLITE. 

This instrument being a combination of Transit and Level, 
its several adjustments are to be made according to the rules 
already given for those instruments. 


CHAPTER XIII. 
EXPLANATION OF TABLES. 


TABLE J.—Contains concise statements of such geometrical 
truths as are applicable to the various discussions in this volume. 
References are given to Davies’ Geometry, in which the demon- 
strations of the propositions may be found. 


TaBLE IJ.—Contains all the formule necessary to the solu- 
tion of any plane triangle; also, a select list of miscellaneous 
formule. A few formule with respect to the versed sine and 
external secant are new. 


TABLE JII.—Contains a complete list of formule expressing 
the relations between the radius, tangent, chord, versed sine, 
external secant, and central angle of a railway curve; also, the 
relations between the radius, degree of curve, length of curve, 
and central angle. The notation is identical with that used 
elsewhere in the book. 


Tas.Ee 1V.—Contains the radius, and its logarithm, for every 
degree of curve to single minutes up’to 10 degrees, and thence 
by larger intervals up to 50 degrees. With the radius is given 
also the perpendicular off-set, ¢, from the tangent to a point on 
the curve at the end of the first 100-foot chord from the tan- 
gent-point, and the midd!e ordinate, m, of a 100-foot chord. 
See eqs. (16, 34, 87, 40, and 805). 


TaspLE V.—Contains the corrections to be added to the tan- 
gents and externals of any railroad curve, as obtained by refe- 
rence to Table VI., according to the degree of the given curve 
(found at head of columns), and its central angle, (found in the 


254 FIELD ENGINEERING. 


first column.) If the given degree of curve, or central angle, 
does not appear in the table, the exact value of the correction | 
may be casily obtained by interpolation. | 


TaBLE VI.—Contains the exact values of the tangents, T, | 
and externals, E, to a1 degree curve, for every 10 minutes of | 
central angle, from 1° to 120° 50’. Approximate values of the 
tangent and external to any other degree of curve may be had 
by simply dividing the tabular values opposite the given cen- 
tral angle by the given degree of curve, expressed in degrees. 
These approximations may be made exact by adding the proper 
corrections taken from Table V. See eqs. (21) and (24). 


| TABLE VII.—Contains the value of Long Chords of from 2 
| ib to 12 stations, for every 10 minutes of degrce of curve from 0° 

| to 15°, and of aless number of stations for degrees of curve be- 
He tween 15° and 80°. As the chord of one station is always 100 
Ht feet, the column of the first station gives instead the length of 
i arc subtended by the chord of 100 feet. See §$121, 122, 128, | 
ia 124, 125. 


TasLE VIII.—Contains the values of Middle Ordinates to 
long chords of from 2 to 12 stations, for every 10 minutes of 
aii degree of curve from 0° to 10°, and of from 2 to 6 stations for 
Hal every curve from 10° to 20°, at 10-minute intervals. The table 

Ml may be used, not only to fix the middle point of an arc, but 
also, in conjunction with the table of long chords, to locate in- 
termediate stations. See §$121, 122, 123, 124, 125. 


TABLE 1 X.—Contains the chords of a scries of angles vary- 
ing by half degrees up to 30° for radii varying by 100 feet up to | 
1000 feet. It shows, therefore, the linear opening between | 
Hal} the extremities of two equal lines at any given number of hun- 

i dred feet from their intersection, when the angle does not ex- 
i) ceed 30°. For any distance exceeding 1000 we have only to 
add to the value found in that column, the value found in the 
i column headed by the excess of distance over 1000 feet. Con- 
versely, the table gives the angular deflection required between 
two equal lines, in order that at a given distance from the point 
of intersection they may be separated a given amount. 


EXPLANATION OF TABLES. 


" : . a 
TABLE X.—1. Contains values of the ratio w= ae accord- 


ing to the notation of $147 for finding the angle 7 (Fig. 34) 
between the radius PO of the curve at any point P, and the 
tangent P& to the valvoid arc PX by the simple formula 
eq. (80)¢ =uwA. The table embraces lengths of curve from 
300 to 2000 feet, and central angles from 10° to 120°. 

LR 4 : : ! 
= 60° w = 4, and for hasty approximation this 
value of w may be assumed in any case without consulting 
the table. 


A F : r : ; 
2. Contains values of the ratio » = 7 for finding the radius 


4 
of the valvoid arc at the point P (Fig. 35) in terms of the 
length of curve £ = AP by the simple formula, eq. (82), 
mol. 

3. Contains values of the length J, of a valvoid are limited 
by two curves of equal length laid out from the same tangent 
and same P.C., but whose central angles differ by 1°. The 
length Z of each curve is given in the first column, and the 

: : eZ ay eee a een 
half sum of their central angles (eee 4) is given at the 
2 
head of the other columns. 

When the central angles of two curves of equal length 
differ by @ degrees the length / of the valvoid arc joining 
their extremities is expressed by the simple formula, Fig. 36, 


eq. (86) Pet Pe PBS CA UAE 
A'+ A‘ 


in which 7, is taken from the column headed by 


fe 
and opposite the given value of Z;or 7 is found by inter- 
polation if necessary. See § 150 and example. 


TABLE XI.—Contains the measurements necessary to lay 
down a turnout with frogs of given numbers or angles for 
both a standard and a three-foot gauge. The distance BF is 
measured on the rail of the given track from the heel of the 
switch to the point of the frog, while af is the chord of the 
centre line of the turnout between the same points. The 
radius 7 applies to the centre line of the turnout. The dis- 


tance af" is measured on the centre line of the straight track 


206 FIELD ENGINEERING. 


from the heel of the switch to the point of the middle frog. 
The length of switch AY should conform to the tabular 
values unless the throw is to be different from that assumed 
in the table. Sce §§ 180, 181, 182. 

TABLE XIJ.—Contains the middle ordinates of chords vary- 
ing in length from 10 to 82 feet, and for degrees of curve vary- 
ing from 1° to 50°. The use of the table is obvious. See § 199. 


TABLE XIiI.—Gives the proper difference in elevation of 
rails on curves of various degrees from 1° to 50° for veloci- 
ties varying from 10 to 60 miles per hour. See § 201. 


Wan TaBLE XIV.—Gives the rise of grades in feet per mile and 
Hi their angle of inclination corresponding to a rise per station 
AAT (100 fect) varying from 0.01 foot to 10 feet. 


i TABLE XV.—Contains values of the formula (log 4 — 1) 
i 60384.3 in which # = reading of the barometer in inches. The 
inches and tenths of the readings are in the left-hand column, 
while the hundredths are found at the top of the other columns. 
The difference of any two values corresponding to two read- 
ings taken simultaneously at any two stations is the differ- 
ence in elevation in feet of those stations. But the differ- 
ence in height so found is subject to a correction for tempera- 
ture given in the next table. See § 10. 


TABLE XV1.—Contains coeflicients of correction for atmos- 
pheric temperature, by which the approximate heights ob- 
tained by Table XY. are to be multiplied for a correction of 
these heights, which correction is to be added or subtracted 
according as the coefficient given in the table is marked 
+ or --. See §11. 


i Tapiu XVII.—Contains corrections in feet, required by the 

i curvature of the earth and the refraction of the atmosphere, to 
be applied to the elevation of a distant object as obtained by a 
Ievel or theodolite observation for distances ranging from 
il 300 feet to 10 miles. See § 119. 


TABLE XVIII.—Contains the cvefficients for reducing the 
space on a vertical rod intercepted by the stadia hairs when 


SXPLANATION OF TABLES. 257 


the line of collimation is inclined to the horizon, to the space 
that would be intercepted were the line of collimation horizon- 
tal; previded, that the visual angle 9 defined by the stadia hairs 
is Such that tan 49 = .005 or 9 = 0° 34’ 22.63, which is its 
customary value in surveying instruments, The angle of in- 
clination @ is taken at every 10 minutes through half a quad- 
rant, 


TABLE XIX,—Contains the logarithms of the coefficients 
given in Table XVIII. 


TABLE XX.—Gives the lengths of circular arcs to a radius 
=a I 

To find the length of any arc expresscd in degrees, minutes, 
and seconds, take from the table the lengths of the given num- 
ber of degrees, minutes, and seconds respecti vely, and multi- 
ply their sum by the length of the radius, The product is the 
length of are required. 


TABLE XXI.—-Contains the values of minutes and seconds 
expresscd in decimals of a degree, for every 10 seconds of arc, 
and also for quarter minutes up to one degree. 


TABLE XXII.—Contains the values of inches and fractions 
expressed in decimals of a foot for every 32d of an inch up to 
one foot. 


TABLE XXIII.—Contains the squares, cubes, square roots, 
cube roots, and reciprocals of numbers from 1 to 1054. Its 
use may be greatly extended by observing that if any number 
is multiplied by 7 its square is multiplied by n’, its cube by 


: ; 1 
n, and its reciprocal by —. 
n 


TABLE XXIV.—The logarithm of a number consists of 
two parts, a whole number called the characteristic, and a dcci- 
mal called the mantissa. All numbers which consist of the 
same figures standing in the same order have the same man- 
tissa, regardless of the position of the decimal point in the 
number, or of the number of ciphers which precede or follow 
the significant figures of the number. The value of the char- 
acteristic depends entirely on the position of the decimal point 
in the number. It is always one less than the number of 


258 FIELD ENGINEERING. 


1 figures in the number to the left of the decimal point. The 
value is therefore diminished by one every time the decimal 
point of the number is removed one place to the Jeft, and vce 
versa. Thus | 


Number. Logarithm. 

13840. 4.141136 

1384.0 3.141136 

188.40 — 2.141186 

13.84 1.141136 

1.384 0.141186 
1384 —1,141136 : 
.01384 —2,141136 | 
.001884 —3,.141136 : 

etc. etc. 


Ht The mantissa is always positive even when the characteristic 
ET is negative. We may avoid the use of a negative characteristic 
| by arbitrarily adding 10, which may be neglected at the close 
i of the calculation. By this rule we have 


Hi Number. Logarithn. 
ie 1.884 0.141136 
ae 1384 9.141136 
. ie .01384 8.141186 
Hy .001884 7.141136 
etc, ete. 


| No confusion need arise from this method in finding a number 
from its logarithm; for although the logarithm 6.141186 repre- 

mil sents either the number 1,384,000, or the decimal .0001384, yet 
these are so diverse in their values that we can never be uncer- 
tain in a given problem which to adopt. 
Halk The table XXIV. contains the mantissas of logarithms, car- 
Wad ried to six places of decimals, for numbers between 1 and 9999, 
} Hi inclusive. The first three figures of a number are given in the 
| first column, the fourth at the top of the other columns. The 
| i first two figures of the mantissa are given only in the second 
we column, but these are understood to.apply to the remaining 
i four figures in either column following, which are comprised 
between the same horizontal lines with the two. 

If a number (after cutting off the ciphers at either end) con- 
i) sists of not more than four figures, the mantissa may be taken 
i direct from the table; but by interpolation the logarithm of a 
number having six figures may be obtained. The last column 
contains the average difference of consecutive logarithms on 


EXPLANATION OF TABLES. 


the same line, but for a given case the difference needs to be 
verified by actual subtraction, at least so far as the last figure 
isconcerned, The lower part of the page contains a complete 
list of differences, with their multiples divided by 10. 


To find the logarithm of a number having six 
figures ;—Take out the mantissa for the four superior places 
directly from the table, and find the difference between this 
mantissa and the next greater in the table. Add to the man- 
tissa taken out the quantity found in the table of proportional 
parts, opposite the difference, and in the column headed by the 
fifth figure of the number; also add #5 the quantity in the col- 
umn headed by the sixth figure. The sum is the mantissa 
required, to which must be prefixed a decimal point and the 
proper characteristic. 

Hrample.—Find the log of 23.4275. 


For 2342 mantissa is 369587 
fevdifls 185 coli7 129.5 
ee ‘é a4 6é 5 9.2, 


Ans. For 23.4275 log is 1.369726 
The decimals of the corrections are added together to deter- 
mine the nearest value of the sixth figure of the mantissa. 


To find the number corresponding to a given 
logarithm.—If the given mantissa is not in the table find the 
one next less, and take out the four figures corresponding to it; 
divide the difference between the two mantissas by the tabu- 
Jar difference in that part of the table, and annex the figures of 
the quoticnt to the four figures already taken out. Finally, 
place the decimal point according to the rule for characteristics, 
prefixing or annexing ciphers if necessary. The division re- 
quired is facilitated by the table of proportional parts, which 
furnishes by inspection the figures of the quotient. 


Example.—Find the number of which the logarithm is 


8.268927 8.263927 
First 4 figures 1836 from 263873 
Diff. 54.0 
Tabular diff. = 236 .. oth fig, = 2 47.2 
6.80 
6th fig. = 3 7.08 


Ans. No. — .0183623 or 188,623,000. 


259 


260 FIELD ENGINEERING. 


reliable beyond the sixth figure. 


table. 


of one second in the arc are given in adjoining columns. 


i angle increases. 
Erample.—Find the log sin of 9° 28’ 20". 
) Log sin of 9° 28’ is 9.216097 
Hit Add correction 20 x 12.62 252 
Ans, 9.216349 
EKxample.—Find the log cot of 9° 28’ 20". 


| Log cotan of 9° 28’ is 10. 777948 
| Subtract correction 20 x 12.97 259 


Ans. 10777689 


The number derived from a six-place logarithm is not 


At the end of table XXIV. is a small table of logarithms of 
numbers from 1 to 100, with the characteristic prefixed, for 
easy reference when the given number does not exceed two 
digits. But the same mantissas may be tound in the larger 


TaBLE XXV.—The logariinmic gine, tangent, 
ete. of an arc is the logaritnin oi the natural sine, tangent, 
etc. of the same arc, but with 10 added to the characteristic to 
avoid negatives. This table gives log sines, tangents, cosines, 

and cotangents for every minute of the quadrant. Wita the 
| number of degrees at the lIefé side of the page are to be read 
Mi the minutes in the left-hand column ; with the degrees on 
| i the right-hand side are to be read the minutes in the right-hand 
i column. When the degrees appear at the top of the page the 
top headings must be observed, when at the bottom those at 
the bottom. Since the values found for arcs in the first quad- 
iH rant are duplicated in the second, the degrees are given from 
0° to 180°. The differences in the logarithms due to a change 


To find the log.sin, cos, tan, or cot of a given 
arc.; Take out from the proper column of the table the log- 
arithm corresponding to the given number of degrees and 
minutes. If there be any seconds multiply them by the ad- 
joining tabular difference, and apply their product as a cor- 

Ai rection to the logarithm already taken out. The correction 1s 
. | to be added if the logarithms of the table are increasing with 
Hii the angle, or subtracted if they are decreasing as the angle in- 
Hit creases. In the first quadrant the log sines and tangents 1n- 
al crease, and the log, cosines and cotangents decrease as the 


EXPLANATION OF TABLES. 


To find the angle or are corresponding to a 
given logarithmic sine, tangent, cosine, or co- 
tangent.—If the given logarithm is found in the proper 
column take out the degrees and minutes directly; if not, find 
the two consecutive logarithms between which the given 
logarithm would fall, and adopt that one which corresponds to 
the least number of minutes; which minutes take out with the 
degrees, and divide the difference between this logarithm and 
the given one by the adjoining tabular difference for a quo- 
tient, which will be the required number of seconds. 

With logarithms to six places of decimals the quotient is 
not reliable beyond the tenth of a second. 


Example. —9.383781 is the log tan of what angle? 
Next less 9.383682 gives 13° 36’ 


Diff. 49.00 -- 9.20 = 05".3 


Ans. 13° 36’ 05".3 


Example, —9.249348 is the log cos of what angle? 
Next greater 583. gives 79° 46' 


Diff. 200 + 11.67 = 20".1 


Ans. 79° 46° 20"1 

The above rules do not apply to the first two pages of this 
table (except for the column headed cosine at top) because 
here the differences vary so rapidly that interpolation made by 
them in the usual way will not give exact results, 

On the first two pages, the first column contains the number 
of seconds for every minute from 1’ to 2°; the minutes are 
given in the second, the log. sin. in the thard, and in the Sourth 
are the last three figures of a logarithm which is the difference 
between the Jog sin and the logarithm of the number of sec. 
onds 1n the first column. The first three figures and the char- 
acteristic of this logarithm are placed, once for all, at the head 
of the column. 

To find the log sin of an are less than 2° given 
to seconds.—Reduce the given arc to seconds, and take the 
Jogarithm of the number of seconds from the table of loga- 
rithms, and add to this the logarithm from the fourth column 
opposite the same number of seconds. The sunr is the log sin 
required. 

The logarithm in the fourth column may need a slight inter. 


262 FIELD ENGINEERING. 
polation of the last figure, to make it correspond closcly to the 
given number of seconds. 


Example.—Find the log sin of 1° 39° 14".4. 


1° 39’ 14".4 = 5954".4 log 3.774838 
add (q —l) 4.685515 


i Ans. log sin 8.460353 


Log tangents of small arcs are found in the same way, only 
taking the last four figures of (¢ — @) from the fifth column. 


Exampile.—Find the log tan of 0° 52’ 35". 


HAGE 52’ 35” = (8120” + 385”) = 8158’ log 3.498999 
Hi add (¢ — l) 4.685609 


| Ans. log tan 8.184608 


i if To find the log cotangent of an angle less than 
ae 2° riven te seconds.—Take from the column headed ( g++ 2) 
it the logarithm corresponding to the given angle, interpolating 
for the last figure 1f necessary, and from this subtract the loga- 
rithm of the number of scconds in the given angle. 


Example.—Find the log cotan of 1° 44° 227.5. 


i 5 qg + ! 15.314292 
i 6240" +- 22".5 = 6262.5 log 3.796748 


| Ans. 11.517544 


aa These two pages may be used in the same way when the 
\ given angle lics between 88° and 92°, or between 178° and 180°; 
} but if the number of degrees be found at the dottom of the paye, 
the title of each column will be found there also; and if the 
. number of degrees be found on the wight hand side of the page, 
ni the number of minutes must be found in the nght hand col- 
umn, and since here the minutes increase upward, the number 
i of seconds on the same line in the first column must be dimin- 
if ished by the odd seconds in the given angle to obtain the num- 
| _ber whose logarithm is to be used with (7+/) taken from the 
. ! table. 


Ezample.—Find the log cos of 88° 41° 12".5 


(¢g — 1) 4.685537 
4740" —12".5 = 4727.5 log 3.674631 


Ans. 8.360168 


EXPLANATION OF TABLES. 


Ezrample.—Find the log tan of 90° 30’ 50”. 
q+ 15.814413 
1800" + 50” = 1850" log 3.267172 
Ans. 12.047241 

To find the are corresponding to a given log 
Sin, cos, tan, or cotan which falls within the 
limits of the first two pages of Tabie XXV. 
Find in the proper column two consecutive logarithms be- 
tween which the given logarithm falls. If the title of the 
given function is found at the top of that column read the 
degrees from the top of the page; if at the bottom read from 
the bottom. 

Find the value of (¢ — ) or (+1), as the case may require, 
corresponding to the given log (interpolating for the last figure 
if necessary). Then if g= given log and / = log of number of 
seconds, 7, in the required arc, we have at once J = g — (q — J) 
or = (¢-+-l) — g, whence n is easily found. 

Find in the first column two consccutive quantities between 
which the number 7 falls, and if the degrees are read from 
the left hand side of the page, adopt the Jess, take out the 
minutes from the second column, and take for the seconds 
the difference between the quantity adopted and the number 
nm. But if the degrees are read from the right hand side of the 
page, adopt the greater quantity, take out the minutes on the 
same line from the right-hand column, and for the seconds 
take the difference between the number adopted and the num- 
ber n. 


Example. —11.734268 is the log cot of what arc? 


qtl 15.314376 
q 1.734268 

n= 3802.8 3.580108 

For 1° adopt 3780. giving 03’ 

Difference 22".8 


Ans. 1° 03' 22".8 or 178° 56’ 37".2. 


Hxrample.—8.201795 is the log cos of what arc? 


7 sil 4.685556 
q 8.201795 
nos 3282".8 d. 916239 

For 89° adopt 3300. giving 05’ 

Difference 17’.2 


Ans. 89° 05' 17".2 or 90° 54’ 42".8. 


264 FIELD ENGINEERING. 


TABLE XXVI.—Contains logarithmic versed sines and ex- 
ternal secants for every minute of the quadrant, with the 
differences of the same corresponding to a change of 1 second 
in the arc or angle. Interpolation for seconds is made in the 
same manner as with log sines of the preceding table, except 
onthe first two pages. For angles less than 4° the differences 
vary so rapidly that interpolation by direct proportion will not 
give exact values. 

On the first two pages the column headed g — 2/ contains 
the difference between the log versed sine (or log ex secant) of 
an are and twice the logarithm of the number of seconds in the 
same are. The characteristic, and first three decimals (9.070) 
are common to all the logaritlims in these columns up to 3° 19’ 
for log vers sines, where it changes to (9.069), as shown at the 
foot of the column; and up to 4° for log ex secants, where it 
changes to (9.071). At the point of change a cipher is replaced 
by the mark ¢ to call attention. 


To find the log vers sin, or log ex sec of an 
angle given to seconds.—Reduce the angle to seconds, 
take the logarithm of this number, multiply it by 2, and add 
the product to the logarithm in the column (q¢ — 2!) found op: 
posite the given angle. The log (g — 2/) should be corrected 
by interpolation for the fractional part of a minute in the given 
angle. 


Hreample.—W hat is the log ex secant of 2° 14’ 48".77 
2° 14’ 43".7 == 8040” + 43.7 = 8083".7 log 3.907610 
9 


QI 7.815220 
(@—2)° 9.070064 
ngs MOM, 6.885284 


To find the are corresponding to a given log 
vers, or log ex sec.—Find in the column of log vers, or 
log ex sec the two values between which the given log falls, 
and take out from the column (qg— 2/) the logarithm corres- 
ponding to the given log (interpolating for the value of the last 
figure if necessary). Subtract this from the given logarithm 
and divide by 2. The quotient is the logarithm of the num- 
ber of seconds in the required are. 


EXPLANATION OF TABLES. 26d 


Example.—7.344728 is the log vers of what arc? 


q 7.344728 
3° 49’ + (q — 21) 9.069960 


2)8.274 274768 
13720".9 ee 4.1373 
13680. 


Ans. 3° 48' 40".9 


To find the log ex sec of an are greater than 
88° given to seconds.—Take from the column (¢-+J) 
the logarithm corresponding to the given arc, interpolating for 
the fraction of aminute. From this subtract the logarithm of 
the number of seconds in the complement of the given arc. 


Hrample.—W hat is the log ex sec of 88° 24' 20".5? 


For 88° 24' g+/ 15.302188 
Correction 129 x a 44 


g+l 15.302227 
Comp. 88° 24’ 20".5 = 5789".5 log 3.758874 


Ans. log ex sec 11.548353 


To find the angie corresponding to a given 
log ex sec when the angle is greater than 88°,— 
Find in the table the two consecutive log ex secants between 
which the given one falls, and then find by interpolation the 
value of the log (¢-+-/) corresponding to the given log ex sec 
and subtract the latter from it. The difference will be the 
logarithm of the number of seconds in the complement of the 
required angle, which is then easily found. 


Hxample.—11.924368 is the log ex sec of what arc? 


Given log ex sec 11.9243868 
Next less 11.918290 g+l 15.809225 
Diff. 6078 
; .  _--- 9852 — 9225 Be ¥4 
Correction = 99141 — 18290 xX 6078 = i1 
q+l — 15.309296 
Given log ex sec 11.924368 
0° 40’ 26".2 = 2426.2 .. log 8.384928 


Ans. 89° 19 83".8, 


266 


FIELD ENGINEERING. 


TABLE XXVII.—Contains natural sines and cosines, to five 


places of decimals for every minute of the quadrant. 


Correc- 


tions for fractions of a minute are made directly proportional 
to the difference of consecutive values in the table; positive 
for sines, negative for cosines. 


TABLE XX VIII.—Contains natural tangents and cotangents 
to five places of decimals for every minute of the quadrant. 
Corrections for fractions of a minute are made directly propor- 
tional to the difference of consecutive values in the table ; 
positive for tangents, negative for cotangents. 


TABLE XXIX.—Contains natural versed sines and external 
secants to five places of decimals for every minute of the 


quadrant. 


Corrections for fractions of a minute are made 


directly proportional to the difference of consecutive values. 
They are positive in every case. 


LAR LB 


Contains the number of cubic yards con- 


tained in prismoids of various side slopes, bases, and depths, 
as indicated by the title and the numbers i. the first column. 
Each prismoid is supposed to have a uniform level cross sec- 


tion throughout. 


These tables are chiefly useful in making up 


preliminary estimates from the profile, or in other cases where 


only approximate results are required. 


For monthly and final 


estimates more elaborate tables are required, such as are des- 


cribed in § 257. 


Oo 
e 


To make an approximate estimate of quanti- 
ties from a profile by use of Table XX X.—Select the 
proper column for base and slopes, and. if the outline of a cut 
on the profile is roughly a four-sided figure, stretch a fine silk 
thread over the surface line to average it, note the depth from 
thread to grade line midway of the cutting, and multiply the 
tabular number opposite this depth by the average length of 


the cutting in stations of 199 fect. 


(By average length is meant 


the half sum of the length of the grade line in the cutting and 
of so much of the surface line as is covered by the thread.) If 
the area of a cutting as seen on the profile is approximately 
triangular, stretch an averaging line over each incline, and 
note the depth from the intersection of these lines to grade, 


and multiply the tabular number opposite this depth by one- 


meen 


EXPLANATION OF TABLES. 267 


half the length of the cut measured on the grade line in sta- 
tions. The resulting quantities will be slightly in excess if the 
ground is level transversely, but-may be too small if the trans- 
verse slope is steep, and cutting on the centre line is small. 
In general they furnish a good approximation. Quantities in 
embankments may, of course, be found similarly. <A cut or 
fill may be divided on the profile into several portions, and the 
contents of each portion found senarately if preferred. 

The content of a prismoid, level transversely, but having 
different end depths, may be found correctly by this table thus: 
add together the quantities opposite each end-depth and 4 times 
the quantity opposite the half sum of the depths; multiply the 
sum by the length in feet, and divide by 600. 


TABLE XX XI.—Contains a variety of useful numbers and 
formule. The logarithms are here given to seven places of 
decimals. 


mN 
ea 
an 
aa 
<q 
Ss 


TABLE I.—GEOMETRICAL PROPOSITIONS. 


The References are to Davies’ Legendre, Revised Edition. 


~I 


(oo) 


9) 


10 | 


11 | 


15 


16 | 


| from the vertex 
of an _ isosceles 
triangle bisects 
the base, 


| If one side of a tri- 


Leo Vi Cor 


He OQ WERE Pome 


eX 


LDL bles ees 


(BTR OYE Bah ees 6 


Vein Conie.. 
NA no G WW IRS oe Satoac 


EEE, Vilise Saeco 


Mio tis). wks | 


IIl., XIV., Cor... 


is 6 Oe 


angle is pro- 
duced, 


| If two triangles are 

mutually equian- 

| gular, 

| If the sides of a 

| polygon are pro- 

| duced in the 
same order, 

|If a figure is a 
quadrilateral, 

If a figure is a 
| parallelogram, 


| : 
If three points are | 
| not in the same | 


| straight line, 
| 
| If two ares are in- 


tercepted on the | 


same circle, 


If two arcs 
similar, 


are 

| If two areas are 
circles, 

If a radius is per- 
chord, 

If a straight line is 


| tangent to .@ 
circle, 


| If from a point 
tangents 
drawn 
the circle, 


pendicular to a | 


to touch | 


| The 


without a circle | 
are | 


| 
1 


No REFERENCE. HYPOTHESES. CONSEQUENCES, 
1” GRD. Ea 5 Ayo Ifatriangleisright | The square on the hypothe- 
angled, nuse is equal to the sum of 
the squares on the other 
two sides. 
2) L, XI, Cor.1....|If a triangle is | It is equiangular. 
| equilateral, | 
ie Bg @ hes eo |If a triangle is | The angles at the base are 
| isosceles, | equal. 
AG TeX ly WOT. ee as. |If a straight line | It bisects the vertical angle. 


And is perpendicular to the 
base. 


| The exterior angle is equal to 


the sum of the two interior 
and opposite angles. 


And their 
sides are 


They are similar. 
corresponding 
proportional. 


of the exterior 
right 


sum 
angles equals four 
angles. 


The sum of the interior angles 
equals four right angles. 


| The opposite sides are equal. 


The opposite. angles are 
equal. It is bisected by its 
diagonal. And its diagonals 
bisect each other. 


A circle may be passed 


through them. 


They are proportional to the 
corresponding angles at the 
centre. 


They are proportional to their 
radii. 

They are proportional to the 
squares on their radii. 


It bisects the chord. And it 
bisects the are subtended 
by the chord. 


It meets itin only one point. 


And it is perpendicular to 


the radius drawn to that 
point. 
There are but two. They are 


equal. And they make 
equal angles with the chord 
joining the tangent points. 


18 


19 


20 


oO 
wt 


25 


TABLE I.—GEOMETRICAL PROPOSITIONS 


aN. 


The References are to Davies’ Legendre, Revised Edition. 


REFERENCE. 


HYPOTHESES. CONSEQUENCES. 


iw en 


DT AVL. 2X 


I1., XVIIL., Cor.2 
eae oa eh, geo 


IV.,XXVIII.,Cor. | 


LV JAA. Coro 


Ve ec LX Cor, 


PV IER: SE 


Eee L Vicaccsensssichoss 


TV Sse Xl Dee ee hd 


If two lines are | They intercept equal ares of 
parallel chords a circle. 

ora tangent and | 

parallel chord, 


If an angle at the |The angle at the circumfer- 


circumference of ence is equal to half the 
a circle is sub-| angle at the centre, 


tended by the 
same are as an 
angle at the cen- | 
tre, 


If an angle is in- | Itisa right angle. 
scribed ina semi- 
circle, 


if an angle is | It is measured by one half of 
formed by a tan- the intercepted are. 
gene and chord, 


If two chords in- | The rectangle of the seg- 
tersect each oth- | ments of the one, equals the 
er in a circle, rectangle of the segments 

of the other. 


And if onechord is | The rectangle of the seg- 
a diameter, and ments of the diameter is 
the other per- equal to the square on half 
pendicular to it, the other chord. ind the 

half chord is a mean pro- 

| portional between the seg- 
ments of the diameter. 


If two secants | The rectangles of each secant 
meet without the and its external segment 
circle, | are equal. 


If a secant and | The rectangle of the secant 

tangent meet, | and its external segment is 
equal to the square on the 
tangent. And the tangent 
is a mean proportional be- 
tween the secant and its 
external segment, 


If a straight line | The sum of the squares on 

from the vertex the two sides is equal to 
of a triangle bi- twice the square ot half the 
secis its base, base increased by twice the 
square of the bisecting line. 


| If a perpendicular | The square of a side opposite 


be drawn from an acute angle is equal to 


the vertex of a the sum of the squares of 
triangle to the | the other :ide and the base, 
base, diminished by twice the 


rectangle of the base and 
the distance from the ver- 
tex of the acute angle to 
the foot of the perpendicu- 


eda 


TABLE II.—TRIGONOMETRIC FORMULAL. 


Let A (Fig. 107) = 


AH => ie 
We then have 


sin A 
cos A 
tan A 
cot A 
sec A 


cosee A 
versin A 
covers A 
exsec A 
coexsec A 
chord A 
chord 2 A 


In the right-angled triangle ABC (Fig. 107) 
het Ao = c, 40 = 0,andtbC = a 
We then have: 


1. sin A == 
De COSTA = 
3. tan A = 
4. cota = 
5.. sec A ce 


6. cosec A = 


8, exsec4 = -- 


9, covers A = 


10. coexsec 4 = 


angle BAC = arc BF, and let the radius AW’ = AB == 


= [39 
= B= 2BC 


vers A oe 


TRIGONOMETRIC FUNCTiONS. 


H_ ik G 


AG 
CRFrasBE 
BK=AL 


Fie. 107. 
= cos B 11. a=csinA=dOtanaA 
=isiy 8 29 b=ccosA=acotaA 
= cot B 13 C= “ = 
=i, C 2 oO. ye E = = == 
sin A cos A 
= tan i ,o3=i\c.cos B= Peo 
= cosec B 15). 2 OeSa6 Si =O taets 
= sec B 48 tel pes : 
IE ‘ = | \Coseb ia sinwe. 
Les COVELS 2 17. &@= V(e+b)(e— b) 
= coexsec B 18 b= ¥(eta)(c—a) 
= versin B 19. ¢ = Vq2 + p2 
= exsec B 20°" C= 90° = A+B 
ab 
QB: = —_— 
rea 3 


ae 


TABLE IIl.—TRIGONOMETRIC FORMULAE. 


Pe ys yee 
SOLUTION OF OBLIQUE TRIANGLES. 
B 
t 
| 
} 
| | 
} 
} it 
} i 1 
ft 
Fie. 108. 1 
| 
ay | i 
| GIVEN. SOUGHT, FORMULA. | 
22 | A, Boa | C, b,c | C = 180° — (A4-B), b= ing gc in B, | 
i 
| i a tf 
| -= ——- sin(A+B li, 
| - slu A. ‘ ni 7) Hi 
| t i 
Rene r sin A : | 
| Bers, O. | BO. es bein ie a . C= 180°—-(A+ B), iW 
} i.) 
| a : | i 
! a iC, ih 
| | sin 4 ; 
| 24 ; Gab |4A+B)|/%(44 B)=9°-—YKC i 
{ | 4 
ae ly, Ka : aes Pa ee 
25 134(A — B)| tang (4 — B) = ——~“ tan 144(A+ B) if 
atb iF 
| 26 i ES A es Ca) ea Ry ha 
ingee ee R \ 
| | | B=K(4+L)—-%U4-=B) 
> 4 Ci OS V4 (u A 1. B) sin V6(A 4 aime B) | 
27 c le = b)- =(a—b 
igs | Tana tare) TS sin gee ie 
i | | } ! 
28 | | area K=Y¥absin C. } 
) ia /(s~b) (s—c) . 
29 | a,b,c A Lets =le(a+b-+e);sinkgA =4/ pies: a i] 
| ’ b) } 
4 
| pate =o: (s- ~b)(s—c) 
i 3 2 A= ; tang A= 
89 | Cos ¥ A= V % /* s (s—a) _ 
a] | a ee 
a8 en 4 eV 3 ts a) (s- b)(s — ec). 
| 31 | |sin A = a : 
| | 2(s— b) (s—c) |} 
| | | vers A = — a 
bc 1] 
32 | area K = Vs(s — a) (8 — 5) (§—oO) | 
| 
ie 407 sin Bio sin C 
| A, B,C,a | area ea we — | 
i 


TABLE IL.—TRIGONOMETRIC FORMULA. 


GENERAL FORMULA, 


41 


2) 
~ 


43 


df 


50 


51 


Or 
ce) 


Si 4 


Sin 4 p= 


sin A= 


cos A = 


COs A= 


cos A = 


1 sin A - : 
tan 4 = —-> eae = 4/ sec? A— 1 
x cot A cos 4 4 
Yb Wakes /1— cos? A 
tad hy Bs pean) Soe ee ae 
/ cos? A cos A 
1— cos? A vers 2.4 
tan A = ae Lope eee 
sin 2 4 sin 2 A 
1 cos A —— 
@ot A => eS te, 4 CcOser Fis 
tan A sin A v 
| 
sin 2 A sin 2 A 1-+ co 
cot Ay =~ = = St 
1 — cos2 # vers 2 4 
tan 4 A 
cot A = — 7 


vers A 
vers A 


exsec A 


cos 4A 


COS 2 x 


1 
cosec A 


2Qsinlkg AcosZA = 


VW % vers2 


exsec A 


= 1—cosA = 


V¥1—cos?A = 


A= ¥Wi6(1—cos2A) 


4-i-—sin?-_ A —= 


sin Atanigd = 


= exsec 4 cos A 


= gsecA-—1 = 


ie mi 
/ vers A 


ves: +e 
att ear oa 


= 2sinAc 


os A 


cos? 4 — sin? A 


vers A cot 4 A 


cot Asin A 


ti 


tan AtankwA = 


tan A cos A 


sin? 14 A 


sin 2 2 


1 cos 2.4 


= exsec Acotlg A 


A 


sin 2 A 


2Qsint7 lg Aa 


{1 = 2 sin! 


TABLE 


II.—TRIGONOMETRIC FORMULZ. 


| 70, 


. tanA-+tan B= 


GENERAL FORMULA. 


6 CE eel 


a i 
53, tan}g A =~ we cited A = cat 4a 
1+ sec A sin 4 
2tan A 
54, 2 = soe 
eres 1 — tan? A 
) 
| sin A 1+ cos A 1 
Raerpeie fe a UE Con Ae 


vers4 sinA  cosec A—cot A 


cot? A —1 
aodacot 2 A= —— 
ae cot 2 Pcot wd 
ly , Mes 
Bi, vers A= — 9 Le Sot f= cog 4 
| 1+ V1—MversA 2+ V2(1+ cos A) 
58. vers 2. A = 2 sin? A =2sin A cos A tan A 
59. exsecl4 A = - ace coe & — 
(1+ cos 4) + ¥2(1 + cos A) 
60. exsec 2. A = _2 tan? 4 
1 — tan? A 
61. sin (4 + B) = sin A.cos B+ sin B.cos A 
| 62. cos CA’ .5) = cos A eos Basin 42sin PB 
63. sin A + sin B = 2sin14(A+ B)cosl4(A — B) 


. sin A —sin B= 2cos(A+ 2)sin'4(4 — B) 
. cos A+ cos B= 2cos (A + B) ccs l4(A — B) 
. cos B— cos A = 2sin 4(A+ DP)sin 4(A — B) 
. sin? ys — sin? B = cos? B — cos? A = sin (A + B) sin (4 — B) 


8. cos? A -— sin? B = cos (A + B) cos (A — DB) 


sin (A+ B) 
cos 4.cos B 


sin (A — B) 


3 4—tan B= 
ae : cos A .cos B 


TABLE III.—CURVE FORMULAE. 


vo 


= 


FORMULA. | 
ve G0 Hae? 2 
foe sin 144 D 
‘ 50 
oO D ee re 
sin 4% R 
A 1 
L = 100 —— 
E D 
Ate LF 
or TOO 
A 
D = 100 — 
” ic 


T=Rtan&a 
C=2R sin’ a 
M=Rvers\% aA 
E= Rexsec 4% a 
R= Tcot’%a 
= Ttanl a 
C=2T cos%a 
M=T cot a.vers%a 


Oo alee 
exsec 4 a 


T= Heot%a 


Cu2p Sw 4 
exsec 4 b 
M= HcosWYa 

C 
2sinlg a 


M=%Ctan a 


Cc 
T = ———~ 
2cos% a 
exsec 4 A 
H-= 1410 ——= = 
sat yi sin lg a 
M 
Prat vers lon 
C=2Mcot%a 
r= yw BBB 
vers 14 A : 
ee ert 
| 


cos % A 


TABLE IJI.—CURVE FORMULA. 


| 
| GIVEN. SOUGHT. 


FORMULA. 


R,T | A 


VT? +R 
C 
2R 


LY (e+ $) Ge 


Ets 


> | 
26 || tanga = 
27 ; 2. [sin W™a = 
| 
| | 
> | ’ | | = 
28 Ire Oe Ts A |sin 4 a = 
{ | | 
oC e | ‘ | aos A 
“| | | cos aA 
| 
| | | 
| H { 
| US ew «See Ses A | vers 4 A 
| | | 
| 
| 
31 | * ; |cos ea = 
| | | 
82 | R, K | A | A 


| 
33 | nS - | cos % 
ye 84 eC a4 A ; cos % 
385 | : | tan 14 


36 T, i ..| A | tan 14 


cos 4 


| cos 4% 
| tan 14 


GC. |c= 


ky 
i 


ae 
Ce 
BR 
Q 
S 


cos 4 4 


| exsec 14 


ray 


tan 4 a 


7s 


97 
4) 


M=R— 


| | VT?2+-R2 


ER 


STO 
/ 2TLE 


- 
Gy A A a ane 
2M 
ie 
C2 — 4M? 
C2+4M? 
M 
ae 
‘wai 
E+M 


~ V(R+% 0) (R—16C) — 


Orv 


8 


v8 


C 


9 
a 


) 


TABLE UI.—CURVE FORMULA 


| 
GIVEN. | SOUGHT. FORMULA. 
| | 
tere : | _RyM@R-M) 
48 R,M | T T = fe ol, 
49 ue | 24 C= 2VM2R —M) 
RM 
4 se | Ph _ == — 
| i eam ae pag iees 
| 51 R, E T T= VEQR+E) 
} | 
9 R V E2R any 
KO 66 1 ee 
fal | . ‘Es R+E 
Bé | ‘ I M= 
53 M RLE 
| er 
id } R = — —_——--—---— a 
if} 54 T C | R Vv 27 4 C) Br C) 
} 
! Jor 
Mei Sie CO 
Mil) 55 fs M |\M=16C 
eaatealti 2 | x 2T4 
i | |  sric 
Het ee | 5 Oy eae oe 
Hg So ‘ E |#=1 ‘4/ STAC. 
Hl | | T+E)(P 
i Bi mle oe ey ie re eed : ne ae 
i | | | ol 22 (r? — Ey | 
es aie 58 6 | Y a / 
ai : be 24 72 
: Hi | E ( jaws Ls E?) 
\ BC “ mi LM == 
59 | M | T2462 
Hi | 
th | M2 14 C)2 
\) 60 C, M ee Ns a) 
Hi 2. AL 
i} c(c2+ 4M?) 
i 61 “ T eens 
| s BBs (C2 — 4M?) 
| 2 
| 62 “ E |= ar 
RO 7 > ee es EM 
| 03 M, Hf R tay = Per ae M 
a | | /E+M 
ii | 64 6é | 7 = E 3 Van 
i il} | | S | V H-M 
vi | | “E+M | 
bal | 65 aS | C CO =2M 4/ Py 
i | | A 2 19 
] e664 Pa. R R3 — FR? seg et + RT? —14 MT2=0 
HM 67 “ E E84. 1? M— ET? -4.MT2=0 
68 “i C 1Cc:+270214M2C0—8M2T=0 
4H2%—C? C7 Wor 
| 28 i pet te eR = = 
69 OF Fe | R R3+ Bh “e R= . 0 
| #0 is fe | 273 — T? C—2 TE?—CE2=0 
| | C2. ©23 
at} M | ee 


| 3 M2H# .- TI ce 
Met B+ Me, j 


279 


TABLE IV.—RADII, LOGARITHMS, OFFSETS, ETC. 


‘oe = eal 


| | ; | 
rid adine | Woga- | Tan. | Mid. || i : Loga- | Tang. | Mid. 
Deg. Ber gheae rithm. | Off. | Ord. || Deg. | Radius. | jithm. Of | Ord. 
; D. R. log. R. t. m. D. R. log. R. t. | m. | 
0° 0’ | Infinite | Infinite | .000 | .000 | 1° 0’) 5729.65 | 3.758128 | .873} .218 - 
1 | 343775. | 5 5386274 | .015 | .001 || 1 | 5685.72 750950 887 | .222 | 
2 | 171887. | 5.235244 | .029 | .007 2 | 5544.83 . 743888 902 | 225 
3 | 114592. | 5.059153 | .044 | .011 3) 5456.82 | 1736939 | .916 | .229 | 
4 | 85943.7 | 4.934214 | .058 | .015 4 | 5371.56 . 730100 931 | .233 
5 | 68754.9 837304 | .073 | .018 5 | 5288.92 (23367 .945 | 236 HW 
6 | 57295.8 | .753123 | .087 | .022 6 | 5208.7 716787 .960 | .240 Ht 
7 | 49110.7 .691176 | .102 | .025 7 | 5181.05 . 710206 974 | .244 | Ht 
8 | 42971.8 .633184 | .116 | .029 8 | 5055.59 708772 .989 | .247 Wii 
9 | 38197.2 | .582031 | .131 | .033 9 | 4982.33 | 697432 | 1.004 | .251 It 
10 | 34377.5 | 4.586274 | .145 | .036 10 | 4911.15 | 3.691183 | 1.018 | .255 | i 
11 | 31252.3 | 4.494881 | .160 | .040 || 11 | 4841.98 | 3.685023 | 1.033 | .258 
12 | 28647.8 .457093 | .175 | .044 || 12 | 4774.7 .678949 | 1.047 | .262 
18 | 26444.2 .422331 | .189 | .047 || 13 | 4709.33 .672959 | 1.062 | .265 i 
14 | 24555.4 | .390146 | .204 | .051 14 | 4645.69 .667051 | 1.076 | .269 i 
15 | 22918.3 |  .360183 | «R18 | .055 15 | 4583.75 .661221 | 1.091 | .273 i 
16 | 21485.9 332154 | .233 | .658 16 | 4523.44 .655469 | 1.105 | .27 | 
17 | 20222.1 . 805825 | .247 | .062 17 | 4464.70 .649792 | 1.120 | .280 i 
18 | 19098.6 -281002 | .262 | .065 18 | 4407.46 .644189 |-1.184 | .284 i 
19 | 18093.4 .257521-| .276 | .069 19 | 4351.67 .638656 | 1.149 | .287 i 
20 | 17188.8 | 4.235244 | .201 | .07 20 | 4297.28 | 3.633194 | 1.164 | .291 i 
21-| 16370.2 | 4.214055 | .3805 | .076 21 | 4244.23 | 3.627799 | 1.178 | .295 i 
22 | 15626.1 .193852 | .320 | .080 2 | 4192.47 .622470 | 1.193 | .298 i 
23 | 14946.7 | .174547 | .335 | .084 23 | 4141.96 .617206 | 1.207 | .302 it 
24 | 14323.6 .156064 | .3849 | .087 || 24 | 4092.66 .612005 | 1.222 | .305 i 
25 | 18751.0 | .138335 | .3864 | .091 || 25 | 4044.51 .606866 | 1.2386 | .309 i 
26 | 13222.1 .121302 | .378 | .095 26 | 3997.49 .601787 | 1.251 | .313 i 
27 | 127382.4 .104911 | .393 | .098 || 27 | 3951.54 .596766 | 1.265 | .316 i 
28 | 12277.7 .089117 | .407 | .102 || 28 | 3906.54 .591803 | 1.280 | .320 
29 |! 11854.3 073877 |. .422 | .105 || 29 | 3862.7 .586K96 | 1.294 | 824 j 
80 | 11459.2 | 4.059154 | .436 | .109 || 30 | 3819.83 | 3.582044 | 1.309 | .327 
31 | 11089.6 | 4.044914 | .451 | .118 || 31 | 8777.85 | 3.577245 | 1.824 | .331 
32 | 10743.0 .031125 | .465 | .116 32 | 3736.7 .572499 | 1.338 | .335 
33 | 10417.5 .017762 | .480 | .120 || 33 | 3696.61 .567804 | 1.353 | .338 
84 | 10111.1 | 4.004797 | .495 | .124 34 | 3657.29 .563160 | 1.367 | .342 | 
85 | 9822.18 | 3.992208 | .509 | .127 | 35 | 8618.80 .508564 | 1.3882 | .345 
36 | 9549.34 979973 | .524 | .181 | 36 | 3581.10 .554017 | 1.3896 | .349 
37 | 9291.29 .968074 | .588 | .135 37 | 3544.19 .549517 | 1.411 | .353 ) 
38 | 9046.75 956493 | .553 | 1138 38 | 3508.02 .545063 | 1.425 | .356 
39 | 8814.7 945212 | .567 | .142 || 39 | 3472.59 .540654 | 1.440 | .360 
40 | 8594.42 | 3.934216 | .582 | .145 4G | 3487.87 | 3.536289 | 1.454 | .364 ig 
41 384.80 | 3.923493 | .596 | .149 i | 3403.83 | 3.531968 | 1.469 | .367 
42 | 8185.16 | .913027 | .611 | .153 42 | 3370.46 .527690 | 1.483 | .3871 
43 | 7994.81 .902808 |. .625 | .156 43 | 3337.7 .523453 | 1.498 | .875 
44 | 7813.11 .892824 |. .640 | .160 || 44 | 3305.65 .519257 | 1.513; .37 
45 | 7639.49 .883065 | .654 | .164 45 | 3274.17 .515101 | 1.527 | .382 
46 | 7473.42 | .873519 | .669 | .167 46 | 8243.25 .510985 | 1.542 | .385 
47 | (314.41 .864179 | .684 | .171 47 | 3212.98 .506908 | 1.556 | .3889 
48 | 7162.03 | .855036 | .698 | .174 48 | 3183.23 502868 | 1.571 | .393 | 
49 | 7015.87 | .846082.| .713 | <178 | 49 | 3154.03 .498866 | 1.585 | .396 
50 | 6875.55 | 3.887308 | .727 | .182 | 50 | 3125.36 | 3.494900 | 1.600 | .400 . 
51 | 6740.74 | 3.828708 | .742 | .185 51 | 3097.20 | 3.490970 | 1.614 | -.404 
52 |#$611.12 .820275 | .756 | .189 52 | 3069.55 487075 | 1.629 | .407 
53 | 6486.38 .812002 | .771 | .198 53 | 3042.39 .483215 | 1.643 | .411 
54 | 6366.26 .803885 | .785 | .196 54 | 3015.7 .479389 | 1.658 | .414 
55 | 6250.51 .795916 | .800 | .200 55 | 2989.48 .475596 | 1.673 | .418 
56 | 6138.90 788091 | .814 | .204 56 | 2963.7 .471836 | 1.687 | .422 
57 | 6031.20 | .780404 | .829 | .207 57 | 2938.39 -468109 | 1.702 | .425 
58 | 5927.22 772851 | .844 | .211 58 | 2913.49 .464413 | 1.716 | .429 ? 
59 | 5826.76 765427 | .B58 | .215 59 | 2889.01 460749 | 1.731 | .433 ! 
60 | 5729.65 | 3.758128 | .873 | .218 60 | 2864.93 | 3.457115 | 1.745 | .4386 


| 2728.52 


Radius. 


R. 


2864.93 
2841.2 
2817. 
2795. 
2772.8 


2750. ¢ 


2707. 
2685. 
2665. 
2644. 
2624. 
2604. 
2584. 9: 
2565. 
2546. € 
2527. 
2509. 
2491.2 
2473.: 
2455.7 
2438, 
2421. 
2404. 
2387. 
2371. 
2854.8 
2338.7) 

2322.98 
2307.39 
2292. 01 


1910.08 


ee 


Loga- 
rithm. 


log. R. 


3 457115 
453511 
449937 
. 446392 
-442876 
-439388 
485928 
.482495 
-429089 
-425710 

3.422356 


3.419029 
415727 
412449 
-409197 


-405968 . 


-402763 
399582 
-896424 
393289 
390176 
3887085 
384016 
- 080969 
3877943 
-874938 
.3(1954 
- 368990 
. 366046 
. 3863122 
860217 
8573382 
354466 
.851618 
348789 
845979 
843187 
.840412 
.887655 
.334916 
3.332193 


3.329488 
326799 
324127 
821471 
318832 
. 316208 
-313600 
-811008 
808431 

3.305869 


3.303323 
-800791 
298274 
-295771 
293283 
290809 


co cw 


Co co 


"288349 


285902 
283470 
8.281051 


WWW*W WWW Ree Prk ek pe ek ee ee ee pe pe 
Jeo) 
se 
Ve) 


0 0 09 09 0 
— 
Or 
© 


2.487 


TH WW WWW WWW 
orore * 
oS 
oO 


: Loga- 
Deg. | Radius. | rithm: 
D. R. log. R. | 
| | | 
| 3° 0’! 1910.08 | 3.281051 
1 | 1899.53 218646 
2 | 1889.09 206253 
3 | 1878.77 278804 
4 | 1868.56 | 271508 | 
5 | 1858.47 | .269155 
6 | 1848.48 | .266814 
7 | 1888.59 - 204486 
8 | 1828.82 .262170 
9 | 1819.14 .259867 
10 | 1809.57 | 3.257576 
11 | 1800.10 | 8.255296 
12 | 1790.7 .253029 
13 | 1781.45 20077 
14 | 1772.27 - 2485380 
15 | 1763.18 246297 
16 | 1754.19 - 244077 
17 | 1745.26 -241867 
18 | 1736.48 239669 
19 | 1727.75 .237481 
20 | 1719.12 | 3.235805 
21 | 1710.56 | 3.233140 
22 | 1702.10 230985 
23 | 1693.75 . 228841 
24 | 1685 .42 .226707 
25 | 1677.2 . 224584 
26 | 1669.06 222472 
27 | 1661.00 - 220369 
28 | 1653.01 218277 
29 | 1645.11 -216195 
80 | 1637.28 | 3.214122 
31 | 1629.52 | 8.212060 
32 | 1621.84 .210007 
33 | 1614.22 - 207964 
34 | 1606.68 . 205930 
85 | 1599.21 . 203906 
86 | 1591.81 .201892 
87 | 1584.48 . 199886 
88 | 1577.21 . 197890 
89 | 1570.01 . 195903 
40 | 1562.88 | 3.193925 
41 | 1555.81 | 8.191956 
42 | 1548.80 . 189996 
43 | 1541.86 . 188045 
44 | 1534.98 . 186103 
45 | 1528.16 . 184169 
46 | 1521.40 182244 
47 | 1514.7 - 180327 
48 | 1508.06 .178419 
49 | 1501.48 .176519 
50 | 1494.95 | 3.174627 
51 | 1488.48 | 8.172744 
52 | 1482.07 . 170868 
53 | 1475.7 . 169001 
54 | 1469.41 . 167142 
55 | 1463.16 . 165291 
56 | 1456.96 .163447 
7 | 1450.81 . 161612 
58 | 1444.72 . 159784 
59 | 1438.68 . 157963 
60 | 1432.69 | 8.156151 


TABLE IV.—RADII, LOGARITHMS, OFFSETS, ETC. 


‘ | 
Tang. | Mid. 


| 
| 


Off. | Ord. | 
tit at 
2.618 | .654 
2.652 | 658 
2.647 | 662 
2.661 | 665 
2.676 | 669 | 
2.690 | .673 
2.705 | .676 
2.7719 | .680 
2.734 | .684 
2.749 | .687 
2.763 | 691 | 
2.778 | .694 
2.792 | .698 
2.807 | .702 
2 821 | .705 
2.836 | .709 
2.850 | .713 
2.865 | .716 
2.879 .| .720 
2.894 | .723 
2. 727 


2. 
2.938 | .734 
2.952 | .738 
2.967 | .742 
2.981 | .745 
2.996 | .749 
3.010 | .%53 
3.025 | .756 
3.039 | .760 
3.054 | .763 
3.068 | .767 
3.083 | .77 


3.127 | |782 
3.141 | 1785 
3.156 | .789 
3.170 | 793 
3.185 | _796 
3.199 | 800 
3.214 | .808 
3.228 | 807 
3.243 | “811 
3.657 | .814 
3.972 | 818 
3.286 | 822 
3.301 | .825 
3316 | _829 
3.330 | _832 | 
3.345 | _836 
3.359 | .840 
3.374 | 848 
3.388 | 1847 
3.403 | 1851 
8.417 | .854 
3.432 | “858 
3.446 | “962 
3.461 | |865 
3.475 | _869 
3.490 | |872 


~~ 


LIL, 


IS ) 3 
TABLE IV,—RADI, LOGARITH 


OFFSETS, ETC. 


WCMOIOor war cS 


Pasa <p Sica iasameente ai aan icbeccs heh ts eoiibiicablaiiaceee rare ME Soe eon a 


5 | 1348, 


03 | 1173.65 


5¢ | 1157.8 


ar by Loga- 
g. _ Radius. rithm. -| 
R. | log. R. | 
1432.69 3.156151 
1426.74 | .154346 ! § 
1420.85 + .152548 
} 1415.01 | .150758 | ¢ 
1409.21 | .148975.) § 
1403.46 , .147200 | : 
1397.76 | .145431 | 
| 1892.10 | .143670 | § 
1386.49 | .141916 
1380.92 | 140170 
| 1875.40 |3,138430 
1369.92 {3.136697 
1364.49 | .184971 
1359.10 | .138251 
1353 .7 . 181539 
45 | 129833 
1843.15 | .128134 
1337.65 | .126442 
1332.77 | .124756 
1327.63 | .123077 
1322.53 |3 121404 
1317.46 |8.119738 
1312.43 | .118078 
1307.45 | .116424 
1302.50 | .114777 
1297.58 | .113136 
1292.7 .111501 
1287.87 | .109872 
1283.07 | .108249 
1278.30 | .106632 
1273.57 |3.105022 
1268.87 |3.103417 
1264.21 | .101818 
1259.58 | .100225 
1254.98 | .098638 
1250.42 | .097057 
1245.89 | .095481 
1241.40 | .093912 
1236.94 | .092347 
1282.51 | .090789 
1228.11 |3.089236 
1223.74 |3.087689 
1219.40 | .086147 
1215.30 | .084610 } 
1210.82 | .083079 
1206.57 | .081553 
1202.36 | .080033 
1198.17 | .078518 
| 1194.01 | .077008 
9 | 1189.88 | .075504 
| 1185.78 |8.074005 
1181.71 |3.072514 
66 | .071022 
| 65 | .069538 
| 1169.66 | .068059 
| 1165.7 . 066585 
1161.76 | .065116 
85 | 063653 
1158.97 } .062194. 
1150.11 | .060740 
1146.28 |3.059290 


ES 


bet et ee hh pp Ph fed fk fe fk feed 


28 


| Mid. || H : ! Loga- | Tan 2 
| Ord. °|| Deg. | Radius. | rithm. Of 
ma. R. | log. R. t. 
872 || 1146 . 28 I3. 059290 | 4.362 
O76 || | 1142.47 | .057846 | 4.276 
. 880 | 1188.69 | .056407 | 4.391 
.883 i 1184.94 | .054972 | 4.405 
.887 | 1181.21 | ,053542 | 4.490 
891 1177.50 | .052116 | 4.435 
894 ' 1123.82 | .050696 | 4.449 
.898 1120.16 | .049280 | 4.464 
. 902 1116.52 | .047868 | 4.478 
. 905 | 1112.91 | .046462 | 4.493 
. 909 1109.83 3: 045059 | 4.507 
.912 | 1105.76 3.043662 4,522 
.916 2 | 1102.22 | .042268 | 4.536 
920 1098.7 .040880 | 4.551 
923 1095.20 | .039495 | 4.565 
927 1091.73 | .038115 | 4.580 
931 | 1088.28 | .036740 | 4.594 
934 1084.85 | .035368 | 4.609 
988 j; 1081.44 | .034002 | 4.623 
942 1078.05 | .032639 | 4.638 
945 1074.68 '3.031281 | 4.653 
.949 1071.34 {3.029927 | 4.667 
952 1068.01 | .028577 | 4.682 
956 1064.71 | .027231 | 4.696 
.960 1061.43 | .025890 | 4.711 
963 | 1058.16 | .024552 | 4.725 
967 1054.92 | .028219 | 4.740 
971 1051.7 .021890 | 4.754 
974 1048.48 | .020565 | 4.769 
978 1045.31 | .019244 | 4.783 
982 1042.14 |3.017927 | 4.798 
985 1039.00 |3.016614 | 4.812 
. 989 1085.87 | .015305 | 4.827 
993 1032.76 | .018999 | 4.841 
. 996 1029.67 | .012698 | 4.856 
.000 1026.60 | .011401 | 4.87 
003 1023.55 | .010107 | 4.885 
007 1020.51 | .008818 | 4.900 
O11 | 1017.49 | .007532 | 4.914 
014 1014.50 | .006250 | 4.929 
.018 1011.51 |8.004972 | 4.943 
22 1008.55 {3.003698 | 4.958 
025 1005.60 | .002427 | 4.972 
029 1002.67 |3.001160 | 4.987 
032 999.762 {2.999897 | 5.001 
036 996.867 | .998637 | 5.016 
.040 993.988 | .997381 | 5.03 
.043 991.126 | .996129 | 5.045 
047 988.280 | .994880 | 5.059 
051 | 985.451 | .993635 | 5.07 
054 | 982.638 |2.992393 | 5.088 
.058 | 979.840 |2.991155 | 5.103 
062 | 977.060 | .989921 | 5.117 
065 | 974.294 | .988690 | 5.132 
.069 | 971.544 | .987463 | 5.146 
073 968.810 | .986238 | 5.161 
076 | 966.091 | .985018 | 5.175 
.080 | 963.387 | .988801 | 5.190 
083 | 960.698 | .9825 87 5.205 
.088 | 958. 025 | .98137 5.219 
091 | 955.366 2: 980170 5.234 


Dora 


TABLE IV.—RADII, 


LOGARITHMS, OFFSETS, ETC. 


283 


; | | 
[eg Ay ‘nee 
Heke Loga- | Tang.| Mid. || : Loga- | Tang.) Mid 
| Deg. | Radius.) yithm. | Off | Ord. | Pes: | Radius. | rithm. | “Off.” | Ord 
| D: R. log. BR. |. | mas |] DL | RL) PogORR. | | me 
| | | | 
G° 0’! 955.366 2.980170 | 5.234 | 1.309 || *7°-0 | 819.020 |2.918295 | 6.105 | 1.528 
1 | 952.722 | .979066 | 5.248 | 1.318 | 1 | 817.077 | .912263 | 6.119 | 1.531 
2 | 950.093 | .977766 | 5.263 | 1.317 2 | 815.144 | .911234 | 6.134 | 1.535 
3 | 947.478 | 976569 | 5.277 | 1.820 3 | 813.238 | .910208 | 6.148 | 1.539 
4 | 944.877 | 975375 | 5.202 | 1.824 || 4 | 811.803 | .909183 | 6.163 | 1.548 
5 | 942.291 | 974185 | 5.206 | 1.327 || 5 | 809.397 | 908162 | 6.177 | 1.546 
6 | 939.719 | .972998 | 5.321 | 1.331 6 | 807.499 | .907142 | 6.192 | 1.550 
7 | 937.161 | .971814 | 5.385 | 1.385 7 | 805.611 | .906125 | 6.206 | 1.553 
8 | 934.616 | .970633 | 5.850 | 1.358 8 | 803.731 | 905111 | 6.221 | 1.557 
9 | 932.086 | 969456 | 5.364 | 1.342 |) —-9 | 801.860 | 904098 | 6.236 | 1.561 
10 | 929.569 |2.968282 | 5.379 | 1.346 || — 10 | 799.997 2.903089 | 6.250 | 1.564 
11 | 927.066 2.967111 | 5.393 | 1.349 || 11 | 798.144 [2.902081 | 6.265 | 1.568 
12 | 924.576 | 965943 | 5.408 | 1.358 || 12 | 796.299 | .901076 | 6.279 | 1.572 
13 | 922.100 | .964778 | 5.422 | 1.356 || 13 | 794.462 | .900073 | 6.204 | 1.575 
14 | 919.637 | .963616 | 5.437 | 1.360 || 14 | 792.634 | .899073 | 6.308 | 1.579 
| 15 | 917.187 | .962458 | 5.451 | 1.364 || 15 | 790.814 | .898074 | 6.823 | 1.582 
a 16 | 914.750 | .961303 | 5.466 | 1.368 16 | 789.003 | .x97078 6.887 | 1.586 
i. 17 | 912.326 | .960150 | 5.480 | 1.37 17 | 787.210 | .896085 | 6.352 | 1-590 
Wei 18 | 909.915 | .959001 | 5.495 | 1.875 || 18 | 785.405 | .895094 | 6.366 | 1.593 
tu 19 | 907.517 | _957855 | 5.510 | 1.378 || 19 | 783.618 | .804105 | 6.881 | 1.597 
Re 20 | 905.131 2.956711 | 5.524 | 1.382 | 20 | 781.840 [2.893118 | 6.395 | 1.600 
i oi | 902.758 2.955571 | 5.539 | 1.386 || 21 | 780.069 |2.892133 | 6.410 | 1.604 
| 92 | 900.397 | .954434 | 5.553 | 1.389 92 | 778.307 | .891151 | 6.424 | 1.608 
Hh, 93 | 898.048 | .953300 | 5.568 | 1.393 23 | 776.552 | .890171 | 6.439 | 1.611 
li 94 | 895.712 | .952168 | 5.582 | 1.397 || 24 | 774.806 | .889193 | 6.453 | 1.615 
ie 95 | 893.388 | 951040 | 5.597 | 1.400 || 25 | 773.067 | .888217 | 6.468 | 1.619 
il 96 | 891.076 | .949915 | 5.611 | 1.404 || 26 | 771.336 | .887244 | 6.482 | 1.628 
| a7 | 888.776 | 948792 | 5.626 | 1.407 27 | 769.613 | .886272 | 6.497 | 1.626 
ih 98 | 886.488 | .947673 | 5.640 | 1.411 28 | 767.897 | .885303 | 6.511 | 1.630 
ial! 99 | 884.211 | .946556 | 5.655 | 1.415 29 | 766.190 | .884336 | 6.526 | 1.638 
Vi 30 | 881.946 2.945442 | 5.669 | 1.418 30 | 764.489 2.888871 | 6.540 | 1.637 
i 31 | 879.693 (2.944331 | 5.684 | 1.422 || 31 | 762.797 |2.882409 | 6.555 | 1.641 
Hil 32 | 877.451 | .943223 | 5.698 | 1.426 || 32 | 761.112 | .881448 | 6.569 | 1.644 
| 33 | 875.221 | .942118 | 5.713 | 1.420 33 | 759.434 | .880490 | 6.584 | 1.648 
| 34 | 873.002 | 941015 | 5.727 | 1.438 || 34. | 757.764 | .879534 | 6.598 | 1.651 
35 | 870.795 | .939916 | 6.742 | 1.437 35 | 756.101 | .878580 | 6.613 | 1.655 
36 | 868.598 | .938819 | 5.756 | 1.440 36 | 754.445 | .877627 | 6.627 | 1.659 
37 | 866.412 | .937725 | 5.771 | 1.444 7 | 752.796 | 876678 | 6.642 | 1.662 
38 | 864.238 | 936633 | 5.785 | 1.447 38 | 751.155 | .8757B0 | 6.656 | 1.666 
39 | 862.075 | .935545 | 5.800 | 1.451 39 | 749.521 | .874784 | 6.671 | 1.67 
40 | 859.922 (2.934459 | 5.814 | 1.455 40 | 747.804 [2.873840 | 6.685 | 1.673 
41 | 857.780 '2.983876 | 5.829 | 1.458 41 | 746,274 |2.872898 | 6.700 | 1.677 
42 | 855.648 | .932295 | 5.844 | 1.462 42 | 744.661 | .871959 | 6.714 | 1.680 
| 43 | 853.527 | .981218 | 5.858 | 1.466 43 | 743.055 | .871021 | 6.729 | 1.684 
44 | 851.417 | .930142 | 5.873 | 1.469 44 | 741.456 | .870086 | 6.743 | 1.68 
45 | 849.317 | .929070 | 5.887 | 1.473 45 | 739.864 | 869152 | 6.758 | 1.601 
| 46 | 847.228 | .928000 | 5.902 | 1.476 46 | 738.979 | .868221 | 6.773 | 1.695 
| 47 | 845.148 | .926933 | 5.916 | 1.480 47 | 736.701 | .867291 | 6.787 | 1.699 
48 | 843.080 | 925869 | 5.931 | 1.484 48 | 735.129 | 1866363 | 6.802 | 1.702 
49 | 841.021 | 924807 | 5.945 | 1.487 49 | 733.564 | 1865438 | 6.816 | 1.706 
50 | 888.972 [2.928747 | 5.960 | 1.491 || 50 | 732.005 [2.864514 | 6.831 | 1.710 
| 51 | 836.933 2.922691 | 5.974 | 1.495 Bi | 730.454 [2.863593 | 6.845 | 1.718 
52 | 834.904 | .921637 | 5.989 | 1.498 || 52 | 728.909 | 862673 | 6.860 | 1.717 
53 | 832.885 | .920585 | 6.003 | 1.502 || 53 | 727.870 | .861755 | 6.874 | 1 720 
54 | 830.876 | .919536 | 6.018 | 1.505 54 | 725.838 | 860840 | 6.889 | 1.72. 
55 | 828.876 | .918489 | 6.082 | 1.510 Bb | 724.312 | .859926 | 6.903 | 1.728 
56 | 896.886 | 917446 | 6.047 | 1.513 || 56 | 722.793 | 859014 | 6.918 | 1.731 
57 ) 824.905 | .916404 | 6.061 | 1.517 BY | 721.280 | /858104 | 6.932 | 1.735 
58 | 822.934 | .915365 | 6.076 | 1.520 || 58 | 719.774 | .857196 | 6.947 | 1.739 
59 | 820.973 | .914329 | 6.090 | 1.524 59 | 718.273 | 1856290 | 6.961 | 1.742 
i “020 |2.913295 | 6.105 | 1.528 60 | 716.779 121855385 | 6.976 | 1.746 


Loga- 
| rithm 


log. R 


TABLE IV.—RADI, LOGARITHMS, OFFSETS, ETC. 


Tang | Mid 
Ord 


m. 


8° 0 | 716.779 |2.855385 
1°} 715.291 “854483 
2 | 713.810 | .853583 | 
3 | 712.335 | “Boek | 
4 | 710.865 | .851787 
5 709.402 | .850892 
6 | 707.945°>| .849999 | 
7 | 706.493 | 849108 | 
8 | 705.048 | .848219 
9 | 703.609 | .847331 | 
: 10 | 702,175 |2.846445 | 
11 | 700.748 |2.845562 
: 12 | 699.326 | .844679 
) 18 | 697.910 | .843799 
: 14 | 696.499 | .842991 
: 15 | 695.095 | .842044 
16 | 693.696 | .841169 
17 | 692.302 | .840296 
: 18 | 690.914 | .839424 
19 | 689.532 | .838555 
: 20 | 688.156 |2.837687 
: 21 | 686.785 |2.836821 
92 | 685.419 -835956 
23 | 684.059 | .83 
24 | 682.704 | 834232 
25 | 681.354 | .83337 
: 26 | 680.010 "9305415 
: 27 | 678.671 | .831G660 
: 28 | 677.338 | .S80805 
: 29 | 676.008 | .829953 
30 | 674.686 /2.829102 
31 | 673.369 |2.828253 
32 | 672.056 | .827405 


83 | 670.748 | 826560 


-825715 


35, | 668.148 | .82487: 
36 | 666.856 | .824032 
37 | 665.568 | .823193 
38 | 664.286 | 829355 
89 | 663.008 | .821519 
40 | 661.736 |2.820685 
41 | 660.468 |2.819852 
42 | 659.205 | .819021 
43 | 657.947 | .818191 
44 | 656.694 | .817363 
45 | 655.446 | 816537 
46 | 654.202 | .815712 
47 | 652.963 | .814889 
48 | 651.729 | .814067 
49 | 650.499 | .813247 


U., $e | 
50 | 649.274 {2.812428 
| 2.811611 


92 | 646.838 | .810796 
53 | 645.627 | .809982 
54 | 644.420 | .809169 
55 | 643.218 | .808358 
56 | 642.021 | .807549 
57 | 640.828 | .806741 
58 | 639.639 | .805935 
59 | 638.455 | .805130 


60 | 687.275 |2.804327 


WNWNWNWWWNWWWd Sip ees oe 


WNWWNWNWNWWWWWD 


ND W W ®W ® ® W 


Tang.| Mid. | Loga- 
On Ord. | Deg. | ‘Radius rithm Oft 
| 
leikte || Gms) ap. || aan.) ietdetes. |cate, 
6.976 | 1.746 | -9° 0'| 637.275 |2.804827 | 7.846 
6.990 | 1.%49 | 1 |-636.099 | .803525 | 7.860 
7.005 | 1.753 | 2 | 634.928 | .802724 | 7.875 
7.019 |1.756 3 | 633.761 | .801926 | 7.889 
7.084 | 1.761 4 | 632.599 | .801128 | 7.904 | 
7.048 | 1.764 | 5 | 631.440 | .800382 | 7.918 
7.063 | 1.768 | 6 | 630.286 | .799538 | 7.933 | 
007 | 1.771 7 | 629.186 | .798745 | 7.947 | 
7.092 | 1.775 8 | 627.991 | .797953 | 7.962 | 
7.106 | 1.778 9 | 626.849 | .797163 | 7.976 
7.121 | 1.782 10 | 625.712 |2.796374 | 7.991 | 
7.135 | 1.786 11 | 624.579 |2.795587 | 8.005 
7.150 | 1.790 12 | 623.450 | .794801 | 8.020 
7.164 | 1.793 | 13 | 622.325 | .794017 | 8.084 
7.179 | 1.797 | 14 | 621.203 | .793234 | 8.049 | 
7.193 | 1.801 15 | 620.087 | .792453 | 8.063 | 
7.208 | 1.804 | 16 | 618.974 | .791673 | 8.078 
7.222 | 1.807 | 17 | 617.865 | .790894 | 8.092 
7.237 | 1.811 | 18 | 616.760 | .790117 | 8.107 | 
7.251 | 1.815 | 19 | 615.660 | .789341 | 8.121 
7.266 | 1.819 | 20 | 614.563 |2.788566 | 8.136 
7.280 | 1.822 | 21 | 613.470 |2.787793 | 8.150 
7.295 | 1.826 | 22 | 612.380 | .787021 | 8.165 | 
7.309 | 1.829 | 23. | 611.295 | .786251 | 8.179 | 
7.824 | 1.833 | 24 | 610.214'| .785482 | 8.194 | 
7.888 | 1.837 | 25 | 609.136 | .784714 | 8.208 | 
7.353 | 1.840 | 26 | 608.062 | .783948 | 8.223 
7.867 | 1.844 27 | 606.992.| .783183 | 8.237 | 
7.882 | 1.848 28 | 605.926 | .782420 | 8.252 
7.896 | 1.851 29 | 604.864 | .781657 | 8.266 
7.411 | 1.855 30 | 603.805 |2.780897 | 8.281 
7.425 | 1.858 31 | 602.750 |2.780137 | 8.295 
7.440 | 1.862 | 32 | 601.698 | .7'79 379 8.310 
7.454 | 1.866 | 33 | 600.651 | .778622 | 8.324 
7.469 | 1.869 34 | 599.607 | .777867 | 8.389 | 
7.483 | 1.873 | 35 | 598.567 | .777112 | 8.353. | 
7.598 | 1.877 | 36 | 597.530 | .776360 | 8.368 | 
7.512 | 1.880 | 37 | 596.497 | .775608 | 8.382 
7.527 | 1.884 | 38 | 595.467 | .774858 | 8.397 
7.541 | 1.887 | 39 | 594.441 | .774109 | 8.411 | 
7.556 | 1.892 40 | 593.419 /2.773361 | 8.426 
7.570 | 1.895 41 | 592.400 |2.772615 | 8.440 
7.585 | 1.899 42 | 591.884 | .771870 | 8.455 
7.599 | 1.908 43 | 590.872 | .771126 | 8.469 
7.614 | 1.906 44 | 589.364 | .770383 | 8.484 
7.628 | 1.910 45 | 588.359 | .769642 | 8.498 
7.643 | 1.914 46 | 587.357 | .768902 | 8.513 
7.657 | 1.918 47 | 586.359 | .768164 | 8.527 
7.672 | 1.921 48 | 585.364 | .767426 | 8.542 
7.686 | 1.924 | 49 | 584.373 | .766690 | 8.556 | 
7.701 | 1.928 50 | 583.385 |2.765955.| 8.571 
7.715 | 1.932 | 51 | 582.400 |2.'765221 | 8.585 
7.780 | 1.935 | 52 | 581.419 | /764489 | 8.600 
7.744 | 1.939 | 53 | 580.441 | .763758 | 8.614 
7.759 | 1.943 | 54 | 579.466 | .763028 | 8.629 | 
7.773 | 1.946 | 55 | 578.494 | .762299 | 8.643 | 
7.788 | 1.950 | 56 | 577.526 | .761572 | 8.658 
7.802 | 1.953 | 57 | 576.561 | .760845 | 8.672 | 
7.817 | 1.957 | 58 | 575.599 | .760120 | 8.687 
7.831 | 1.961 || 59 | 574.641.| {759307 | 8.701 
7.846 | 1.965 || 60 | 573.686 |2.758674 | 8.716 


ROS Ff AE ee eat bet et 


G 


WwW 


WW NWWWYD 


965 


968 
9G . 
SON 

: 9479 
.983 
. 987 
.990 
994. 
.998 
.001 
.005 
.008 


.012 
.016 
019 
023 
.026 
030 
034 
037 
041 
045 


.048 
.052 


056 


.060 
.063 
.066 
070 
O%4 


077 
081 
084 
.088 
.092 
.096 
.099 
.108 
.106 
.110 


113 
S17 


121 


125 
.128 
.182 
oe 


Radius. 


TABLE IV.—RADII, LOGARITHMS, OFFSETS, ETC. 


Loga- 


562.466 
10.638 


3.823 
’,019 


52227 
3.447 


678 
920 
174 
3.438 
714 


3.001 


.298 


39. 606 
37.924 | 


).253 
593 


32.943 


803 


29.673 
28.053 
26.443 


843 


3.252 


671 
100 
589 
).986 


5.443 
3.909 
2.385 


.869 
.863 


JT. 865 


376 
.896 


3.425 


. 962 
507 
061 


97.624 


5.195 


hin 
odd 


3. 801 
956 


.559 
SAW 
’.790 
5.417 


5 051 
33.694 
344 


.001 
.666 


60 | 478.839 


“) 


() 


ww 


rithm. 
R. log. R. 
3.686 |2.75867 
(1.784 | .757282 
39.896 | . 755796 
38.020 | .754364 
6.156 | .752937 
44.305 | .751514 

750096 


4274 
745870 
144471 
.743076 
741686 
.'740300 
738918 
737541 
736169 
734800 
738436 
732077 
. 730721 
. 729870 
. 728023 
. 726681 
(253842 
- 724008 


722677 


(21351 
720029 
18711 
2.717397 
(16087 
“14781 
713479 
712181 
710887 
109596 
708310 
107027 
105748 
704473 
103202 
701934 
100671 
699410 
698154 
696901 
695652 
694407 
693165 


691926 
.690692 


WWW WNW WWNWWW 


WWW WWWWWWYW 


a WWW 


. 689460 
. 688283 
. 687008 
685788 
. 684570 
683357 
682146 
. 680989 


'2.679735 


Radius. 


R. 


442.814 


441 . 684 
440.559 


| 489.440 


438.326 
437.219 
436.117 
435.020 
433.929 
432.844 
431.764 
430.690 
429. 620 
428.557 
427.498 
426.445 
425.396 
424.354 
423.316 
422.283 
421.256 
420.233 
419.215 
418.203 
417.195 


| 416.192 


415.194 
414.201 
413.212 
412.229 
411.250 


| 410.275 


rw) 


co] 


a 


.676145 |10.! 
674954 |10U. 
673767 |10.! 
672584 |10.62 
.671403 |10.6: 
670226 |10.6 
2.669052 |10.71: 


2.667881 |10. 
666713 |10. 
665549 |10. 

| .664387 10. 
663229 |10. 

662074 |10.8 

660922 |10.5 

659773. |10.4 

658628 |10. 

657485 |11. 

656345 

655208 |11. 

654075 |11. 

652044 

651816 |11. 

650691 |11. 

649570 |11. 

648451 |11. 

647835 |11. 

646221 /11. 

645111 |11. 

644004 |11. 

642899 |11. 

641798 |11. 

640699 |11. 

639603 |11. 

638510 |11. 

637419 |11. 

636331 |11. 

635246 |11. 

634164 |11. 

633085 |11.6 

682008 |11.6 

630934 |11.6 

629863 |11 

628794 |11.7 

627728 |11. 

626665 |11.£ 

625604 |11.£ 

624546 11.8 

2.623490 11. 

622437 |11. 

621387 |11 

620889 |11 

619294 

618251 12 

617211 |12. 

616173 12 

615138 |12 

614106 |12 

613075 |12 


= 
eo 


| 
—" 


Loga- | Tang. | 


rithm. Off. 
log. R. | t. 
9 |2.679735 |10.453 
| 678535 (10.482 | 
.677388 |10.511 


. 956 
985 
014 


.043 | & 


O71 
100 
129 
158 
187 


3% *~— 


9% ~~ 


~ WW 


WWW WWNHWNYW WW WW WNWWNWWYW 
0 OO = See aS spedis 


927 | 2.$ 


EEE EEO a ee 


Loga- 
rithm, 


log. R. 


275 |2.613075 


- 612048 
.611028 
.610000 
. 608980 
.607962 
. 606946 
605933 
.604923 
.603914 


. 602908 


57| .601905 | 


609904 
-999905 
.598908 
.097914 


.596922 | 


.995933 
594945 


.593960 | 


592978 


-991997 
.991019 
.990043 
.589069 
988097 
.987128 | 12 
.086161 | 


985196 


984233 


2.58327: 
“582314 


-5813858 
.580403. | 
.579451 


978501 
977553 
.976608 
.975664 


944722 


.573783 


972845 
.971910 


|) .070977 
970045 


.569116 
. 568189 


| .567264 

566340 
565419 
564500 


.563582 
.562667 
.561754 
. 560843 
559933 
.559026 
.558120 
.557216 
.556315 
2.555415 


ic=>) 
Co Oo 8 GO OO 2 9 


Coon cu co CD Co CD 
De 


Loga- 
‘| rithm, 
log. R. 


2.555415 
004517 
.553621 
.552727 
.551834 
550944 
590055 
.549169 
548284 
547401 


2.546519 
545640 
544762 
543887 
543013 
542140 

752 | .541270 

540401 

| 539585 

| 588670 

537806 

| 536945 

| .B36085 

535227 

584370 

533516 

532663 

531811 

530962 

530114 


|2.529268 
528424 
527581 
526740 
| .525900 
525062 
524226 
523392 
BR255$ 
521728 
2.520898 
520070 
519244 
518419 
517596 
516774 
515954 
| 515136 
| .514319 
| 513504 


.512690 
5| .511878 
.511067 
| .510258 
509451 
5 | .508645 
9} .507840| 
| 507037 
506236 
.505436 


| 319.623 2.504638) 12 


13.917 
18.946 
13.975 
14.004 
14,033 
14.061 
14,090 
14.119 


14,148 | & 


14.177 
14.205 
14.234 
14.263 
14.292 
14.320 
14.349 
14.378 
14.407 
14.436 
14.464 


14.493 
14.522 
14.551 
14.580 
14.608 
14.637 
14.666 
14.695 
14.723 
14.752 
14.781 
14.810 
14.838 
14.867 
14.896 
14.925 
14.954 
14.982 
15.011 
15.040 


15.069 
15.097 
15.126 
15.155 
15.184 
15.212 
15.241 
15.270 


ae 
oO 


re) 
e 


60 | 287. 


939 2.459300 


TABLE IV.—RADII, LOGARITHMS, OFFSETS, ETC. 


: Loga- 
Radius. rithm. 

R. log. R. 

| 319.623 |2.504638 
319.037 | .503841 
318.453 | .503045 
317.871 | .502251 
317.292 | .501459 
316.715 | .500668 
316.139 | .499879 
315.566 | .499091 
314.993 | .498304 
314.426 | .497519 
313.860 |2.496736 
313.295 | .495953 
312.732 | .495173 
312.172 | .494393 
311.613 | .493616 
311.056 | .492839 
310.502 | .492064 
309.949 | .491291 
309.399 | .490518 

| 308.850 | .489748 
308.303 |2.488978 
307.759 | .488210 

1 | 307.216} .487444 

> | 306.675 | .486679 
306.136 | .485915 
305.599 | .485152 

| 305.064 | .484391 
304.531 | .483632 

6 | 304.000 | .482873 

| 303.470) .482116 

‘| 302.943 |2.481361 

| 302.417 | .480607 

| 301.893 | .479854 
301.371 | .479102 

| 300.851 | .478352 

| 300.833 | .477603 

| 299.816 | .476855 

| 299.302 | .476109 

| 298.789 | .475364 
298 .278| .474621 

| 297.768 |2.473878 

| 297.260 | .473137 
296.755 | .472398 

| 296.250 .471659 

| 295.748 | .470922 

| 295.247 | 470186 

2 | 294.748 | .469452 
34 | 294.251] .468718 
3 | 203.756 | .467986 

| 998.262 | .467256 

| 292.770 |2.466526 

42 | 292.279| .465798 
44 | 291.790} .465071 
291.803 | .464345 

48 | 290.818| .463621 
50 | 290.384 | .462897 
52 | 289.851} .462175 
| 289.371 | .461455 
288 .892| .460735 

58 | 288.414| .460017 


—_ 


Pe ek ek ek ek ek ek Pk Rk 


| BRRQVQVRMIAINVWWIRN NRO 


Mid. 
Ord. | 


| 4.089 
| 4.096 
| 4.103 
eT 
118 | 
AP} 
4.188 || 
4.140 | 
147 || 
5155.1) 
162 || 
.169 
sa leerg 
184 | 
191 || 
.199 || 
206 


9 | 4.352 || 
| 4.360 || 
4.367 || 
365 | 4.3874 || 830° 0/| 193.185 


| 287. 
285. 
| 283. 
30 | 280. 
| 278. 
| 276: 
| 274. 
272. 
270. 
| 268. 
| 266. 
0) | 264. 


262. 
260. 
| 258. 
| 256. 
254. 
252. 
| 250. 
| 249. 
| 247. 
30 | 245. 
40 | 243 
50 | 242. 


| 24° 0’| 240. 
10 | 238 

20 | 237 

30 | 235. 

40 | 234 

50 | 232. 
25° 0’| 231 
10 | 229 

20 | 228 

30 | 226. 

40 | 225. 

50 | 2238 

| 26° 0’ 222. 
10 | 220. 
20 | 219. 
30 | 218. 
40 | 216 
50 | 215. 
|| 27° 0’| 214. 
10 | 212. 
20 | 211. 
30 | 210. 
40 | 209. 
50 | 207. 
| 206. 
10 | 205 
20 | 204. 
30 | 208. 
40 | 201. 
50 | 200. 
| 29° 0’, 199. 
10 | 198. 

20 | 197. 
30 | 196. 
40 | 195. 
50 | 194. 


. |Radius. 


R. 


939 | 
583 
267 
988 
746 
541 
370 
234 
132 
062 
024 
018 


042 
098 
180 
292 
431 
599 
793 
013 
258 


529 


825 


144 
487 | 


853 
241 


652 


084|. 


537 


O11 |2 
506 
020 | 


4-4-4 


555 
108 


.680 
271 2. 


879 
506 
150 


811 


489 
183 
893 
620 
362 
119 
891 


78 | 


480 
296 
125 | 
969 
826 
696 
580 
476 
B85 
306 
240 


Crore Ore 


C2 > SD OVOT ON OT OL OT OTOH 


> DD DADAIADAHI AIS 
or or E 


Deg. 


D. 


Radius. 


TABLE IV.—RADI, 


Loga- | Tang. 
rithm, | 


R. |log R.| t. 


~ 
~~ 
fore 
ans 


40 


30° 20’) 191.111 2.281286 | 26.163 
189.083 | .276652. | 26.443 


: 31° (| 187.099} 272071 | 26.724 


32° 


: 20 


4) 


20 | 185.158! .267541 | 27:004 
183.258} .268062 | 27.284 
0’ | 181.898 | .258632 : 
179.577 | .254250 | 27.843 

40 | 177.794 .249916 | 28.123 
88° 0/| 176.047 | 245628 | 28.462 


\) 
~ 
on 
ror) 
nse 


20 | 174.336] .241386 | 28.680 
0 | 172.659] 287188 | 28.959 


4 
0! | 171.015 
169.404] .228924 | 99. 

167.825 | .224855 | 29.793 


233035 | 29.237 


ci) 


~~] 
oS 
Or 
pm, 
on 


0’| 166.275 | .220828 | 30.07 


164.756 | .216842 | 30.348 
163.266 | .212895 | 80.625. 


0’ | 161.803 |} .208988 | 30.902 


160.368 | .205119 | 31.178 
158.960 | .201288 | 31.454 


Q’| 157.577 | .197494 | 31.730 


156.220 | .193736 | 32 006 
154.887 | .190014 | 32.282 


0’ | 153.578 |2.186328 | 82.557 


TABLE V.—CORRECTIONS FOR TANGENTS AND EXTERNALS. 


LOGARITHMS, OFFSETS, ETC. 


2. ae 42.262 


| 2 
Mid. se |Radina | Loga- | Tang. | Mid. ; 
Ord. | Deg. Radius. rithm. Of | Ord. | 

m. || D. R log. R.| t. m. 
| 

| 6.657 || 38°30’) 151.657/2.180863 32.969 | 8.479 
6.731 || 39° 0’) 149.787] .175475 33.381 | 8.592 
6.805 || 30 | 147.965) .170160 33.792! 8.704 
6.879 || 40° 0’| 146.190) .164918 34.202! 8.816 
6.958 || 144.460) .159747 34.612] 8.929 
7.027 || 41° 0’, 142.778] .154645 35.021! 9.041 
7.101 || 141.127) .149610 35.429) 9.154 
7.175 || 42° 0’ 139.521] .144641) 35.887 | 9.967 
7.250 30 | 187.955} .189736 36.244] 9.380 
7.324 || 43° 0’ 136.425] .134895 36.650! 9.493 
4.398 134.932) .130114 37.056! 9.606 
7.473 || 44° 0’ 133.478/2 .125395 37.461 | 9.719 
7.547 30 | 182.049} .120734 37.865 | 9.832 
7.621 || 45° 0’ 130.656} .116130 38.268] 9.946 
7.696 30 | 129.296) .111584 38.671 110.059 
7.770 || 46° 0’ 127.965} .107092 39.073 }10.173 
7.845 80 126.664} .102655 39.474 |10.286 
7.919 || 4'7° 0’ 125.392 -098270 39.875 |10.400 
7.994 30 124.148) .093938, 40.275 110.516 
8 068 || 48° 0’ 122.930} .089657, 40.674 110.628 
8.148 30. 121.738} .085425. 41.072 110.742 
8.218 || 49° 0% 120.571) .081243 41,469 110.856 
8.292 30 | 119.429} .077109, 41.866 |10.970 
8.367 || 50° 0’, 118.310 11.085 


For TANGENTS, ADD 
10° | 15° | 20° | 25° | goe ||Ang 
Cur.| Cur.| Cur.| Cur.| Cur.|| © 
06 Oe 1S Peas 19 || 10° 
13 19| .26| .82 39 || 20 
19 .29| .89) .49] .59]} 30 
26 40 Bia 12267 80)! 40 
.o4 .51/ .68] .85/)1.021) 50 
.42 .63| .84/ 1.05 11.271! 60 
.51 76 | 1.02 | 1.28 | 1.541) 7 
61 .91 | 1.22 | 1.53) 1.84]| 80 
72 | 1.09 | 1.45 | 1.83 | 2.20]! 90 
.86 | 1.380) 1.74 | 2.18 | 2.62 \100 
1.03 | 1.56 | 2 08 | 2.61 | 3.141 /110 
1.25 | 1.93 | 2.52 | 3.16 | 3.81 ||120 


For EXTERNALS, ADD 


5° | 10° 
Cur.| Cur. 

001 | .003 
.006 | .011 
.013 | .025 
.023 | .046 
.087 | .075 
.056 | .112 
.080 | .159 
.110 | .220 
.149 | .299 
.200 | .401 
.268 | .536 
.360 | .721 


15° 

| Cur. 

004 
O17 
"038 


290° 
Cur. 


006 


25° 


007 
028 
.065 
ers 
.189 
.283 
-403 
.558 
756 
1.015 
)1.355 
\1.825 


Cur. | 


f 


TABLE VIL—TANGENTS AND EXTERNALS TO A 1° CURVE. 


| | | | 
S Tan- | Exter- || Tan- | Exter- || Tan- | Exter- 
Angle. gent nal. | Angle. gent nal. | Angle. gent. | nal, 
i T. E.— 4+ 2 T. E. A To) Ble 
1° 50.09 218 || 11 551.70 | 26.500 || Qy°¢ 1061.9 | 97.57% 
10’| 58.34 297 10’| 560.11 | 27.813 || 10’| 1070.6 | 99.155 
20 | 66.67 .888 20 | 568.53 | 28.137 || 20 | 1079.2 | 100.75 
30 | 75.01 491 30 | 576.95 | 28.974 || 30 | 1087.8 | 102.35 
40 | 83.84 .606 40 | 585.36 | 29.824 || 40 | 1096.4 | 103.97 
50 | 91.68 733 50 | 593.79 | 30.686 50 | 1105.1 | 105.60 
2 100.01 873 || 12 602.21 | 31.561 | 99 1118.7 | 107.24 
19 | 108.35 | 1.024 10 | 610.64 | 32.447 || 10 | 1122.4 | 108.90 
20 | 116.68 | 1.188 20 | 619.07 | 33.347 90 | 1181.0 | 110.57 
30 | 125.02 | 1.864 30 | 627.50 | 34.259 30 | 1139.7 | 112.25 
40 | 133.36 | 1.552 40 | 635.93 | 35.188 40 | 1148.4 | 113.95 
50 | 141.7 1.752 50 | 644.37 | 36.120 || 50 | 1157.0 | 115.66 
3 150.04 | 1.964 || 13 652.81 | 37.070 || 93 1165.7 | 117.38 
10 | 158.38 | 2.188 10 | 661.25-! 38.031 || 10 | 1174.4 | 119.12 
90 | 166.72 | 2.425 20 | 669.70 | 39.006 || 20 | 1183.1 | 120.87 
30 | 175.06 | 2.674 30 78.15 | 39.993 || 30 | 1191.8 | 122.63 
40 | 183.40 | 2.934 40 | 686.60 | 40.992 40 | 1200.5 | 124.41 
50 | 191.7 3.207 50 | 695.06 | 42.004 | 50 | 1209.2 | 126.20 
4 200.08 | 3.492 || 14 703.51 | 43.029 | 94 1217.9 | 128.00 
10 | 208.43 | 3.790 10 | 711.97 | 44.066 || 10 | 1226.6 | 129.82 
20 | 216.77 | 4:099 20 | 720.44 | 45.116 | 20 | 1235.3 | 131.65 
30 | 225.12 | 4.421 30 | 728.90 | 46.178 30 | 1244.0 | 183.50 
40 | 233.47 | 4.755 40 | %87.37 | 47.253 40 | 1252.8 | 185.35 
50 | 241.81 | 5.100 50 | 745.85 | 48.841 || 50 | 1261.5 | 137.23 
5 950.16 | 5.459 || 15 754.32 | 49.441 || 95 1270.2 | 189.11 
10 | 258.51 | 5.829 10 | 62.80 | 50.554 | 10 | 1279.0 | 141.01 
20 | 266.86 | 6.211 20 | 771.99 | 51.679 || 20 | 1287.7 | 142.93 
30 | 275.21 | 6.606 30 | 779.77 | 52.818 || 30 | 1296.5 | 144.85 
40 | 283.57 | 7.018 40 | 788.26 | 53.969 40 | 1305.3 | 146.79 
50 | 291.92 | 7.482 50 | 796.75 | 55.132 || 50 | 1314.0 | 148.7 
6 300.28 | 7.863 || 16 805.25 | 56.309 | 96 1822.8 | 150.71 
10 | 308.64 | 8.807 | 10 | 818.75 | 57.498 || 10 | 1331.6 | 152.69 
20 | 316.99 | 8.762 || 20 | 822.25 | 58.699 || 20 | 1340.4 | 154.69 
30 | 825.35 | 9.230 30 | 830.76 | 59.914 | 30 | 1849.2 | 156.7 
40 | 333.7 9.710 40 | 839.27 | 61.141 || 40 | 1858.0 | 158.72 
50 | 342.08 | 10.202 || 50 | 847.78 | 62.381 | 50 | 1366.8 | 160.76 
uf 350.44 | 10.707 || 17 856.80 | 63.634 | 27 1375.6 | 162.81 
10 | 358.81 | 11.224 10 | 864.82 | 64.900 || 10 | 1884.4 | 164.86 
20. |.367.17 | 11.753 20 | 873.35 | 66.178 || 20 |.1393.2 | 166.95 
30 | 375.54 | 12.294 30 | 881.88 | 67.470 | 30.| 1402.0 | 169.04 
40 | 383.91 | 12.847 40 | 890.41 | 68.774 || 40 | 1410.9 | 171.15 
50 | 392.28 | 13.413 50 | 898.95 | 70.091 || 50.| 1419.7 | 173.27 
8 400.66 | 13.991 || 18 907.49 | 71.421 || 28 1428.6 | 175.41 
10 | 409.03 | 14.582 10 | 916.03 | 72.764 | 10 | 1437.4 | 177.55 
20 | 417.41 | 15.184 20 | 924.58 | 74.119 | 20.| 1446.3 | 179.72 
30 | 425.79 | 15.799 30 |. 933.18 | 75.488 | 30 | 1455.1 | 181.89 
40 | 434.17 | 16.426 40 | 941.69 | 76.869 | 40 | 1464.0 | 184.08 
50 | 442.55 | 17.065 50 | 950.25 | 78.264 || 50 | 1472.9 | 186.29 
9 450.98 | 17.717 || 19 958.81 | 79.671 || 29 1481.8 | 188.51 
10 | 459.32 | 18.381 || 10 | 967.38 | 81.092 || 10 | 1490.7 | 190.7 
20 | 467.71 | 19.058 20 | 975.96 | 82.525 || 20 | 1499.6 | 192.99 
30 | 476.10 | 19.746 || 30 | 984.53 | 83.972 || 30 | 1508.5 | 195.25 
40 | 484.49 | 20.447 || 40 | 993.12 | 85.431 | 40 | 1517.4 | 197.53 
50 | 492.88 | 21.161 50 | 1001.7 | 86.904 | 50 | 1526.3 | 199.82 
10 501.28 | 21.887 || 20 / 1010.3 | 88.389 || 30 1535.3 | 202.12 
10 | 509.68 | 22.624 10 | 1018.9 | 89.888 || 10 | 1544.2 | 204.44 
20 | 518.08 | 23.375 20 | 1027.5 | 91.399 20 | 1553.1 | 206.77 
30 | 526.48°| 24.188 |! 30 | 1036.1 | 92.924 30 | 1562.1 | 209.12 
40 | 534.89 | 24.918 || 40 | 1044.7 | 94.462 | 40 | 1571.0 | 211.48 
543.29 | 25.700 || 50 | 1053.3 | 96.0138 50 | 1580.0 | 213.86 


TABLE VI.—TANGENTS AND EXTERNALS TO A 1° CURVE. 


‘pes ee ha ee ee ee ae 
Tan- | Exter- | Tan- | Exter- Tan- | Exter- 
Angle. gent nal. {| 4ngle.|- cont. | nal. || Angele. gent. | nal. 
T. KE. A T. E. A ae EK. 
: 3ie 1589,0 | 216.25 || 41° | 2142.2 | 387.38 || 51° | 2732.91 618.39 
| 10 | 1598.0 | 218.66 10’) 2151.7. | 390.71 10’| 2748.1 622.81 
20 : 1606.9 | 221.08 | 20 | 2161.2 | 394.06 | 20 | 2753.4 627.24 
: 30 | 1615.9 | 223.51 | 30 | 2170.8 | 897.43 | 380 | 2763.7 631.69 
: 40 | 1624.9 | 225.96 40 | 2180.3 | 400.82 40 | 2773.9 636.17 
: 50 | 1633.9 | 228.42 | 50.| 2189.9 | 404.22 | 50. | 2784.2 640.66 
: 32 1643.0 | 230.90 || 42 2199.4 | 407.64 52 2794.5 645.17 
| 10 | 1652.0 | 233.39 | 10 | 2209.0 | 411.07 | 10 | 2804.9 649.7 
: 20 | 1661.0 | 235.90 | 20 | 2218.6 | 414.52 20. | 2815.2 654.25 
: 30 | 1670.0 | 238.43 80 | 2228.1 | 417.99 30 | 2825.6 658 .83 
: 40-| 1679.1 | 240.96 | 40 | 2237.7 | 421.48 40 | 2885.9 663 .42 
: 50 | 1688.1 | 243.52 | 50 | 2247.3 | 424.98 50 | 2846.3 668 .03 
| 33 1697.2 | 246.08 || 43 2257.0. | 428.50 || 53 2856.7 | 672.66 
: 10 | 1706.3 | 248.66 10 | 2266.6 | 432.04 10 | 2867.1 677.382 
: 20: | 171523: | 251.26 20. | 2276.2 | 485.59 20 | 2877.5 681.99 
: 30 | 1724.4 | 253.87 30. | 2285.9 | 439.16 30 | 2888.0 686.68 
40 | 1733.5 | 256.50 40 | 2295.6 | 442.95 40 | 2898.4 691.40 
50 | 1742.6 | 259.14 | 50. | 2805.2 | 446.35 | 50 | 2908.9 696.13 
: 34 1751.7 | 261.80 || 44 2314.9 | 449.98 || 54 2919.4 700.89 
) 10 | 1760.8 | 264.47 | 10 | 2324.6 | 458.62 10 | 2929.9 705.66 
| 20 770.0 | 267.16 | 20 | 2334.3 | 457.27 | 20 | 2940.4 710.46 
: 30-1 1779.1 | 269.86 80 | 2344.1 | 460.95 30 | 2951.0 715.28 
40 | 1788.2 | 272.58 | 40 | 2353.8 | 464.64 40 | 2961.5 720.11 
50 | 1797.4 | 275.31 50 | 2363.5 | 468 .35 50 | 2972.1 (24.97 
35 1806.6 | 278.05 | 45 2373.3 | 472.08 || 55 2982.7 729.85 
10 | 1815.7 | 280.82 10 | 2883.1 | 475.82 10 | 2993.38 734.76 
20 | 1824.9 | 283.60 20 | 2392.8 | 479.59 20 | 8003.9 739.68 
30,! 1834.1 | 286.39 30 | 2402.6 | 483.37 30 | 3014.5 744.62 
40 | 1843.3 | 289.20 40 | 2412.4 | 487.17 40 | 3025.2 749.59 
50 | 1852.5 | 292.02 50 | 2422.3 | 490.98 50 | 3035.8 754.57 
36 1861.7 | 294.86 || 46 2482.1 | 494.82 || 56 3046.5 759.58 
10 | 1870.9 | 297.72 | 10 | 2441.9 | 498.67 10 | 3057.2 764.61 
| an 2 1880.1 | 300.59 | 20 | 2451.8 | 502.54 20 | 8067.9 769.66 
: 30 | 1889.4 | 803.47 | 30 | 2461.7 | 506.42 80 | 8078.7 | 774.78 
40 | 1898.6 | 306 .37 40 | 2471.5 | 510.33 40 | 3089.4 779.83 
50 | 1907.9 | 309.29 50 | 2481.4 | 514.25 50 | 3100.2 | 784.94 
37 1917.1 | 312.22 || 47 2491.3 | 518.20 || 57 8110.9 790.08 
10 | 1926.4 | 815.17 10 | 2501.2 | 522.16 | 10 | 3121.7 | 795.24 
20 | 1935.7 | 318.18 20 | 2511.2 | 526.13 20 | 3182.6 800.42 
30 | 1945.0 | 821.11 30. | 2521.1, | 530.18 | 30 | 3148.4 | 805.62 
40 | 1954.3 | 824.11 40 | 2531.1 | 584.15 | 40 | 3154.2 | 810.85 
50 | 1963.6 | 327.12 50. | 2541.0 | 5388.18 50 | 8165.1 | 816.10 
38 | 1972.9 | 380.15 || 48 2551.0 | 542.23 || 58 3176.0 821.37 
10 | 1982.2 | 333.19 10 | 2561.0 | 546.30 10 | 4186.9 826.66 
20 | 1991.5 | 336.25 20 | 2571.0 | 550.39 20 | 3197.8 831.98 
30 | 2000.9 | 339.32 30 | 2581.0 | 554.50 30 | 3208.8 | 837.31 
40 | 2010.2 | 342.41 40 | 2591.1 | /58.63 40 | 8219.7 | 842.67 
50 | 2019.6 | 345.52 50 | 2601.1 | 562.77 50 | 8280.7 848.06 
39 2029.0 | 348.64 || 49 2611.2 | 566.94 || 59 8241.7 853.46 
10 | 2038.4 | 351.78 10. | 2621.2 | 571.12 10 | 8252.7 858.89 
20 | 2047.8 | 354.94 20 | 2631.3 | 575.32 20 | 3263.7 864.34 
30 | 2057.2 | 358.11 | 30 | 2641.4 | 579.54 30 | 8274.8 869.82 
| 40 | 2066.6 | 361.29 | 40 | 2651.5 | 583.78 | 40 | 3285.8 875.82 
50 | 2076.0 | 364.50 | 50 | 2661.6 | 588.04 | 50 | 3296.9 880.84 
40 2085.4 | 367.72 || 50 2671.8 | 592.32 || 60 3308.0 886.38 
10 | 2094.9 | 370.95 10 | 2681.9 | 596.62 10 | 3319.1 891.95 
20 | 2104.3 | 374.20 20. | 2692.1 | 600.93 | 20 | 3330.3 897.54 
{ 80 | 2118.8 | 877.47 | 3 2702.3 | 605.27 80 | 3341.4 908.15 
| 40 | 2123.3 | 380.76 | 40 | 2712.5 | 609.62 | 40 | 3352.6 908 . 7 
50 | 2132.7 | 384.06 50 | 2722.7 | 614.00 50 | 3363.3 914.45 
ey 


290 


TABLE VIL—TANGENTS AND EXTERNALS TO A 1° CURVE. 


Tan- 
gent. 


4 ke 


3375.0 


3386.3 
3397.5 
3408.8 
3420.1 
3431.4 
3442.7 
3454.1 
3465.4 
3476.8 
3488 .3 
3499.7 
3511.1 
3522. 
3584 
3545. 
3557. 
3568. 
3580. 
3591. 
| 8603 
3615 
3626. 
3638. 


SOR RICO] NOH MOWIWAHM 


1002.3 
1008.3 
1014.4 
1020.5 
1026.6 
1082.8 
1039.0 
1045.2 
1051.4 
1057.7 


1063.9 
1070. 
1076. 
1082. 
1089. 
1095. 
1102. 
1108. 
1115: 
1121. 
1128 
1134. 
1141 
1148 
1154. 
1161. 
1168. 
1174. 
1181 
1188 
1195 
1202. 
1208. 
1215. 


1222. 
1229 
1286. 
1243. 
1250. 
1257. 
1265. 
1272 
127 

1286. 
1293.6 
1300.9 


DOW PODHWIOH DWVHARWIWOARW 


orp et OO CO-IVIWIAVI’ WD 


920.14 
925 .85 
931.58 
937.34 
943.12 
948 . 92 
954.75 
960.60 

966 .48 

972.38 

978.31 
984.27 | 
990.24 | 
996 .24 


| 1890. 


| 1413. 


COO WHO MWWIAIH OUCH QS ~~ 


| 1583. 


i 
DO wWIRADOWUDWA CHOROCUR RAH 


eo 


WDNWOVAP www 


1308 .2 
1315.6 
1822. 
1330. 
1337 
1345 
1352. 
1360. 
1367. 
1375. 
1882. 


1398. 
1405. 


1421. 
1429. 
14236. 
1444. 
1452. 
1460. 
1468. 
1476. 
1484. 
1492. 
1500. 
1508 
1516. 
1524. 
1533. 
1541 
1549. 
1558. 
1566. 
1574. 


1591. 
1600. 
1608. 
1617. 
1625 
1634. 
1643. 
1651 
1660. 
1669 
1678 
1686. 


DH WAIORH 


| 


SOHOHUWOS 


rt Sort Od MRI WONRH ORNATE PP POTHOOW IOS HOC 


fa 
aS; 
co) 
aE 
JIN DW 


Tan- | Exter- 
gent. nal. 
| se EK. 
4893.6 | 1805.3 
10'| 4908.0 | 1814.7 
20 | 4923.5 | 1824.1 
30 | 4937.0 | 1833.6 
40 | 4951.5 | 1843.1 
50 | 4966.1 | 1852.6 
4980.7 | 1862.2 
10 | 4995.4 | 1871.8 
20 | 5010.0 | 1881.5 
30 | 5024.8 | 1891.2 
40 | 5039.5 | 1900.9 
50 | 5054.38 | 1910.7 
| 5069.2 | 1920.5 
10 | 5084.0 | 1980.4 
20 | 5099.0 | 1940.3 
30 | 5113.9 | 1950.3 
40 | 5128.9 | 1960.2 
50 | 5143.9 | 1970.3 
| 5159.0 | 1980.4 
10 ; 5174.1 | 1990.5 
20 | 5189.3 | 2000.6 
30 | 5204.4 | 2010.8 
40 | 5219.7 | 2021.1 
50 | 5284.9 | 2081.4 
| 5250.3 | 2041.7 
10 | 5265.6 | 2052.1 
20 | 5281.0 | 2062.5 
30 | 5296.4 | 2073.0 
40 | 5811.9 | 2088.5 
50 | 5327.4 | 2094.1 
| 5843.0 | 2104.7 
10 | 5858.6 | 2115.3 
20 | 5874.2 | 2126.0 
30 | 5389.9 | 2136.7 
40 | 5405.6 | 2147.5 
50 | 5421.4 | 2158.4 
| 5437.2 | 2169. 
10 | 5453.1 | 2180. 
20 | 5469.0 | 2191 
0 | 5484.9 | 2202. 
“40 | 5500.9 | 22138. 
50 | 5517.0 | 2224. 
5533.1 | 2235. 
10 | 5549.2 | 2246. 
20 | 5565.4 | 2258. 
80 | 5581.6 | 2269 
40 | 5597.8 | 2280 
50 | 5614.2 | 2292 
5630.5 | 2303. 
10 | 5646.9 | 2315. 
20 | 5663.4 | 2326. 
30 | 5679.9 | 2388. 
40 | 5696.4 | 2849. 
50 | 5718.0 | 2361 
| 5729.7 | 2378. 
10 | 5746.3 | 2885. 
20 | 5763.1 | 23897. 
30 | 577! 2408. 
40 | 5796.7 | 2420. 
50 | 5813.6 | 2482. 


SDD OM WN DNAOM CHWONMWNMNWH WE 


TABLE VI.—TANGENTS AND EXTERNALS TO A 1° CURVE 


i | | | 
Tan- |. Bx | | or | | 
| Angle, : 2s Angle Tan- Ex- | ¥ x 
| ent. | ternal. ngle. | oo iter Ste pa eee 
| | ‘ | Be | 2 | | gent. | ternal. | Angle. | gent. .| ternal 
P| ics ale eee dn aedeinsl ick, ae e e e ie E. 
| ; ‘ 
: 91° | 5830.5 | | | | 3 
5830.5 | 2444.9 | ° | 6950.6 | 3978 
10’| 5847.5 | 3457 ’ on 10’) pate | 3278.1 || L11° | 8336.7 | 4386.1 
Ba idooe ee Bab? tH 971.3 | 3204-1 || ~~ 10"| 8362.7 | 4407.6 
20 | 5864.6 | 2469.3 || 20 | 6992 | 99 pose 4407.6 
| gies ee et 20 | 6092.0 | 3310.1 20 | 8388.9 | 4429.2 
| 40 | 5898.8 | 2493.8 || 30 | Role. | 8326.1 | 30 | 8415.1 | 4450.9 
i | 50 | 5916.0 | 2506.1 || 50 | t05d'5 | aces | Boe oe mele 
: 99 | 5933.2 | 95185 || 102 eign 3358.5 50 | 8468.0 | 4494.6 
| | 10 | 5950.5 | 2531.0 || 10 | 7006:6 | B301°2 || 22? 49 | SA94-8 | 4518.6 
| | ~~ 20 | 5967.9 | 2543'5 || 20 | 11718 | sare Se eer = aoa 
ih | ~~ 80} 5985.3 | 9556.0. 80] ri39.0 | Sakae Site 
= Sapien 30 | 7139.0 | 3424.3 30 | 8575.0 | 4583.4 
| sy | 0002-7 | 2568.6 |) | 7160.3 | 3440.9 40 | 8602.1 | 4606 
6020.2 ROSL.o | 50 181.7 | 8457 @ J 6.0 
eens | (181.7 | 3457.6 50 | 8629.3 | 4628.6 
93 6037.8 | 2594.0 || 198 7203.2 | ar | 
| 10 | 6055.4 | 2606'8 || ~~ 10 | waee:7 | 3401-3 |] 228 49 | 8638-6 | 4651-8 
| 20 | 6073.1 | 2619-7 | oat eerie) aco 10 | 8684.0 | 4674.2 
| 30 | 6090.8 | 2632'6 | 30 | 796801 sree’ || a0 | 8711.5 | 4697.2 
| 40 | 6108.6 | 2645 5 40 | 2080°8 | 8025.2 || 380 | 8739.2 | 4720.3 
| Ey ised Pee. | 40 | 7289.8 | 3542.4 || 40 | 8767.0 | 4743.6 
7 94 6144.3 | 2671.6 || 104 ae | 8559.6 50 | 8794.9 | 4766.9 
| 10 | 6162.2 | 9684.7 | 10 Ente | 8576.8 | 114 8822.9 | 4790.4 
| 20 | 6180.2 | 2697.9 || 20 | 7377.8 | 3611.7 Si eee ae 
| 30 | 6198.3 | 2711.2 | BOT anarad Seed eed Seen 
: 40 | 6216.4 | ovoq's | 30 | 7299-9 | 3629.2 30 | 8907.7 | 4861.7 
| 50 | 6234.6 | 2737.9 || 50 | vaaa'e | soau'e cabernet sce 
Maree ose e tart gdiace’ | | dots tel PO seEPOS. 0: 20002 
10 | Ga7i.i | areas || 105 ie ie 3682.3 || 115 8993.8 | 4934.1 
| 20 | 6289.4 | 9779's || ee eS 9.6 | 3700.2 || 10 | 9022.7 | 4958.6 
: 30\16307.0 | ste's 20 | 512.2 | 8718.2 20 | 9051.7 | 4983.1 
| 40 | 6326.3 | 2805.6 40 | 1587-7 | Bea 0 Wi eaie | cae 
: 50-| 6344.8 | S819 4 il Perse 3754.4 || 40 | 9110.3 | 5032.6 
96 | 6363.4 | 9833 9 | 106 7603.5 | 3072.6 | 50 9139.8 5057.6 
| 10 | 6382.1 | 2847.0 | 10 | 7626-6 | song. || 248 44 | $183-4 | p08: 
at mea toe 4 || 10 | 7626.6 | 3809.4 | 10 | 9199.1 | 5107.9 
Seba eee 3 il 20 | 7649.7 | 3827.9 20 | 9229.0 | 5133.3 
40 | 6438.4 | 2889°0 ye cone ea he eugeeere ai eS 
siesta: 40 | (606.3 | 3865.2 40 | 9289.2 | 5184.5 
A nae 50 | 7719.7 | 3884.0 | 50 | 9319.5 | 5210.3 
: 10 | 6495.2 | 29316 |; 107 10 (143.2 | 3902.9 || 117 9349.9 | 5236.2 
| Gipsy rare asian 10 | 7766.8 | 3921.9 10 | 9380.5 | 5262.3 
30 | 6533.4 | 2960.3 30 ‘i483 oo Sn otis shee 
40 | 6552.6 | der4'2 | ee Ay 2060.1 30 | 9442.2 | 5315.0 
50 | 6571.9 | 2989.2 50 | va6e.1 | swe. Fo ieesne ¢ heer 
96 5012 | sees.8 || 10g.” | seed | 2998:7 50 | 9504.4 | 5368.2 
: 10 | 6610.6 | 3018.4 10 | 7910.4 | doar || 228, | 9535-7 | 5895.1 
20 | 6630.1 | 3033"1 90 | 7934°6 | aoe s 10 | 9567.2 | 5422.1 
30 | 6649.6 | 3047.9 30 | 7950.0 | dory'a 30 pegs ee 
gree a ge 830 | 7959.0 | 4077.2 30 | 9630.7 | 5476.5 
sotltene eee S| 40 | 7983.5 | 4097 1 40 | 9662.6 | 5504.0 
90 ane ea 7 117.0 50 | 9694.7 | 5531.7 
10 | 6728/4 | 310777 || 29° 40) B04 | ater a | te ty hoes | no 
20 | 6748.2 | 3122.9 | 20 | 8082.3 | 4178 20 | 9ro20 | bee's 
ay bres 4 t cee | 0 | S082. 4177-6 20 | 9792.0 | 5615.5 
40 | 6788.1 | 3153.3 | 40 8130°3 met: fe see cpteonde 
50 | 6808.2 | 3168.7 || | Seeeeeeml oon See 9857.7 | 5672.3 
LOO hecae stares lags 50 8157.5 | 4239.0 50 | 9890.8 | 5700.9 
0 | 6848.5 | 3199.6 || ~~ 10 | g208/9 gaoo-7 |, 120 peed 0 | 6120.7 
eee | | 10 | $208.2 | 4280.5 | 10 | 9957.5 | 5758.6 
Big hee A 20 | 8288.7 | 4901-4 20 | 9991.0 | 5787.7 
Doe. | 30 | 8259.3 | 4302.4 30 |10025.0 | 5817.0 
50 | 6930.1 | 3262.3 50 | 8310.8 | 4960°¢ at dee ae Me Ee 
| 262.3 | 50 | 8310.8 | 4364.8 50 |10093.0 | 5876.1 


TABLE 


VII.—LONG CHORDS. 


i= el 
Actual Lone CHORDS. 
Degree | _ Are, 
of One | 
eee 2 | 3 A. 5 6 
CoE . Station. Stations. Stations. | Stations. | Stations. | Stations. 
| | 
O° 10’ 100.000 200.000 299.999 399.998 499.996 599.993 
20 000 199.999 299.997 399.992 499 . 983. 599.970 
30 000 199.998 299 992 399.981 499 . 962 599.933 
40 001 199.997 299.986 399.966 499 . 932 599. 882 
50 001 199.995 299.979 399.947 499 . 894 599.815 
1 100.001 199.992 299.970 399.924 499.848 599.733 
10 002 199.990 299.959 399.896 499.793 599.637 
20 002 199.986 299 .946 399.865 499 .729 599.526 
30 003 199.983 299 . 932 399.829 499.657 599.401 
40 003 199.979 299.915 899.789 499.577 599.260 
50 .004 199.974 299 .898 399.744 499.488 599.105 
2 100.005 199.970 299 .878 399.695 499.391 598 .934 
10 006 199.964 299.857 399.643 499 285 598.750 
20 007 199.959 299.834 3998586 499.171 598.550 
30 008 199.952 299.810 399 524 499 .049 598.336 
40 009 199.946 299.783 399 .459 498.918 598.106 
50 .010 199.939 299.756 399.389 498.778 597.862 
3 100.011 199.931 299.726 399.315 498 . 630 597.604 
10 S013 199 . 924 299 .695 399 . 237 498 .474 597.33 
20 014 199.915 299.662 399.154 498 .309 597.043 
| 30 015 199.907 299 .627 399 .068 498.136 596.740 
40 “017 199.898 299.591 398.977 497 .955 596.423 
50 019 199.888 299.553 398 . 887, | 497.765 596.091 
4 100.026 199.878 299.513 398.782 497 .566 595.7 
10 022 199.868 299.471 398 .679 497.360 595.383 
20 .024 199.857 299 428 398.571 497.145 595.007 
30 | -026 199.846 299.388 | 398.459 | 496.921 594.617 
40 028 199.834 299 .337 398.343 | 496.689 594.212 
50 030 199.822 299 .289 398 .223 496 .449 593.792 
5 100.032 199.810 299 .239 398.099 | 496.201 593.358 
10 034 199.797 299.187 397 .970 495.944 592.909 
20 .036 199.783 299 134 397.837 495.678 592.446 
30 .038 199.770 299.079 397.700 495.405 591.968 
40 041 199.756 299 .023 397.559 495.123 591.476 
50 043 199.741 298.964 397.418 494 832 590.97 
6 100.046 199.726 298.904 397 .264 494.534 590.449 
10 .048 199.710 298 843 397.110 494 227 589.913 
20 051 199.695 298.77 396.952 493.912 589.364. 
30 054 199.678 298.714 396.790 493 588 588 .800 
40 .056 199.662 298 .648 396 .623 493.257 588 221 
50 059 199.644 298.579 396 .453 492.917 587.628 
ui 100.062 199.627 298.509 396.278 492.568 587.021 
10 065 199.609 298 .438 396 .099 A92 212 586.400 
20 068 199.591 298 . 364 395.916 491.847 585.765 
30 071 199.57 298 .289 395.729 491.474 585.115 
40 075 199.553 298 .212 395 .538 491 .093 584.451 
50 078 199.533 298.134 395 .342 490.704 583.773 
8 100.081 199.513 298 . 054 395.142 490.306 583.081 
10 .085 199.492 297 .972 394.938 489 .900 582.375 
20 088 199.471 297 .888 394.731 489 .486 581.654 
30 092 199.450 297 803 394.518 489 .064 580.920 
40 095 199.428 297 .'716 394.302 488 . 634 580.172 
50 099 199.406 297 .628 394 .082 488.196 | 579.409 
b 9 100.103 199.383 297.538 393.857 487 .749 578.633 
10 107 199.360 297 .446 398.629 87 294. 577.843 
20 A111 199.337 297 352 393.396 486. 832 577.039 
30 115 199.313 297 257 393.159 486 .361 576 . 222 
40 119 199.289 297.160 392.918 485.882 575.390 
50 123 199.264 297 .062 392.678 485.395 54.545 
10 100.127 199.239 296.962 | 892.424 484.900 573.686 


293 


sca tea tS SEL EO 


Degree 


of 


Curve. 


O° 


10 


10/ 
20 
30 
40 


TABLE VII.—LONG CHORDS. 


Lone CHoRDSs. 


8 


vi 9 10 11 12 

Stations. | Stations, | Stations. | Stations. | Stations. | Stations. 
699. 988 799. 982 899.974 999. 965 1099.95 1199.94 
699 .953 799.929 899.899 999.860 1099.81 1199.76 
699.893 799.840 899.772 999. 686 1099.58 1199.46 
699.810 799.716 899.594 999.442 1099.25 1199.08 
699.704 799.556 899.365 999. 128 1098 . 84 1198.49 
699.574 799 .360 899.086 998.744 1098 .33 1197 .82 
699. 420 799.180 898.757 998 . 290 1097.72 1f97 .04 
699.242 798 . 863 898.376 997. 768 1097 .02 1196.13 
699.041 798.562 897.945 | 997.175 1096 .23 1195.11 
698.816 798 .224 897 .464 996.518 1095.35 1193.96 
698 .567 797 .852 896.931 995.782 | 1094.38 1192.69 
698 .295 797 444 896.349 994. 981 1093.31 1191.31 
698 . 000 797 000 895.716 994.112 1092.15 1189.80 
697.680 196 .522 895.033 993.17 1090.90 1188.18 
697.338 796.008 894.299 992.165 1089.56 1186.43 
696.971 195.459 893.515 991.088 1088.12 1184.57 
696.581 794.874 892.681 989.943 1086.60 1182.59 
696.168 794.255 891.798 988 . 729 1084.98 1180.49 
695.731 793. 600 890.864 987.447 1083.28 1178.28 
695.271 792.911 889.880 986 096 1081.48 1175.94 
694.787 792.186 888 . 846 984.677 1079.59 1173.49 
694.280 791.427 887.763 | 983.190 1077.61 1170.93 
693. 750 790. 632 886.630 | 981.636 1075.54 1168.25 
693.196 789 .803 885.448 980.014 1073.3 1165.45 
692.619 788 . 939 884.217 978.325 1071.14 1162.54 
692.018 788 .040 882.936 976.569 1068.81 1159.51 
691.395 787.108 881 . 606 974.746 1066.38 1156.37 
690.748 786 . 140 880 . 228 972.856 1063.87 1153.12 
690.079 785.188 878.800 970. 900 1061.27 1149.7 
689.386 784.101 77 824 968.877 1058.59 1146.28 
688 . 670 783 .030 75.800 966.788 1055.81 1142.69 
687.930 781 . 925 874.227 964. 634 1052.95 1138.99 
687.169 780.786 872.605 962.415 1050.01 1135.18 
686 . 384 779.612 870.936 960.130 1046.97 1131.26 
685.576 778.406 869.219 957.780 1045.86 1127.24 
684.745 77.165 867.454 955 . 366 1040.66 1123.10 
683.892 775.890 865.642 952.888 1037.37 1118.86 
683 .016 774.582 863.782 950.345 1034.01 1114.51 
682.117 773.240 861.875 947.739 1080.55 1110.05 
681.195 771.864 859. 922 945.069 1027.02 1105.49 
680.251 770.455 857.921 942.337 1023.40 1100.83 
679.285 769.014 855.874 939.542 1019.7 1096.06 
678 296° 767 .539 853.780 936 . 684 1015.93 1091.19 
677 .284 766.03 851.640 933 . 764 1012.07 1086 .22 
676.250 764.490 849.455 930.783 1008.13 1081 .15 
675.194 762.916 347 224 27.741 1004.11 1075.98 
674.116 761.309 844.947 924.638 1000.01 1070.71 
673.015 759.670 842.625 921.47 995.834 | 1065.34 
671.892 757.999 840.258 918.250 991.580 | 1059.88 
670.748 756.295 837.845 914.966 987.250 | 1054.82 
669.581 754.560 835.389 911.623 982.844 | 1048.66 
668 .393 752.792 832.888 908.221 978.362 | 1042.91 
667 .182 750 . 993 830.342 904.761 973.806 | 1037.06 
665.950 749.161 827.754 901 . 242 969.175 | 1031.13 
664.697 747.299 825.121 897 66% 964.471 1025.11 
663.421 745.404 822.445 894 .033 959.694 | 1018.99 
662.124 743.479 819.726 890.343 954.844 1012.79 
660.806 TAL 522 816.965 886 .597 949.924 | 1006.49 
659. 466 739.585 814.160 882.795 944 , 933 1000.12 
658.105 737 68 811.314 878.938 939.871 993.653 


204 


eee re 


Curve. 


10° 10’ 
20 


11 


12 


13 


14 


15 


16; 


17 


18 


19 


20 


Actual 
Are, 
One 

Station. 


100.131 
136 


100.183 


Lona CHORDS, 


TABLE VII.—LONG CHORDS. 


2 
Stations. 


199.213 
199.187 
199.161 
199.134 
199.107 
199.079 
199.051 
199.023 
198 . 994 
198.964 
198.935 


198.904 
198.874 
198.843 
198.811 
198.779 
198.747 
198.714 
198.681 
198 .648 
198.614 
198.579 
198 .544 
198.509 
198.474 
198.437 
198.401 
198.364 
198.327 
198 .289 
198 .251 
198.212 
198.173 
198.134 
198.094 


198.054 
198.013 
197.972 
197.930 
197.888 
197.846 
197.803 
197.760 
197.716 
197.672 
197.628 
197.583 
197.538 
197.492 
197.446 
197.399 
197.352 
197.305 
197.256 
197.209 
197.160 
197.111 
197.062 
197.012 
196.962 


3 | 4 5 
Stations. | Stations. | Stations. 
296.860 392.171 484.397 
296.756 391.914 483.886 
296.651 391.652 483 .367 
296.544 391.387 482.840 
296 .436 891.117 482.305 
296 825 390.843 481 .'762 
296.214 390.565 481.211 
296.100 390.284 480.6538 
295 .985 389.998 480.086 
295.868 389.708 479 511 
295 .750 389.414 478 .929 
295 .629 389.116 478.838 
295 .508 888.814 77.740 
295 . 384 888 .508 77.135 
295 .259 388.197 476 521 
295 .182 387.883 475 .899 
295 .004 887.565 475.27 
294.874. 387 243 474.633 
294.742 386 .916 473.988 
294 .609 386.586 473.336 
294 474 386 .252 472.675 
294 337 885.914 472.007 
294.199 885.572 471 .382 
294.059 385 . 225 470.649 
293.918 884.875 469.958 
293.77. 384.521 469 .260 
293 .629 884.163 468 .554 
293 .483 883.801 467.840 
293 3835 383 .435 467.119 
293.185 383.065 466.3890 
293 .034. 382.691 465 . 654 
292.881 882.313 464.911 
292 .'726 881.931 464.160 
292.57 381.546 463.401 
292.412 881.156 462.635 
292 . 252 380.763 461.862 
292.091 880.365 461.081 
291.928 79.964 460.293 
291 . 764 79.559 459.498 
291.598 379.150 458 .695 
291.430 378.737 457.886 
291.261 378.3820 457.069 
291.090 377.900 456.244 
290.918 3877 47 455.413 
290.743 77.047 454.57. 
290.568 376.615 453.728 
290.390 876.179 452.875 
290: 211 875.739 452.015 
290.031 875.295 451.147 
289 .849 874.848 450.373 
289.665 374.397 449 392 
289.479 373.942 448 504 
289 292 373.483 447.608 
289 . 104 373.021 446.706 
288 .913 372 554 445.797 
288 .722 872.084 444.881 
288 .528 71.610 443 957 
288 .3833 371.1383 443 .028 
288 . 137 370.652 442.091 
287 .939 370.167 441.147 


295 


6 
Stations, 


572.818 

71.926 
571.027 
570.113 
569.186 
568.245 
567.292 
566.3824 
565.343 
564.349 
563.341 


562.321 
561.287 
560.240 
559.180 
558.107 
557.020 
555.921 
554 °809 
553.684 
552.546 
551.395 
550.232 


549.056 
547.867 
546.666 
545.452 
544.226 
542.987 
541.736 
540.472 
539.196 
537.908 
536.608 
535.296 
583.972 
582.635 
531.287 
529.927 
528.555 
af eal 

525.77 

524.369 
522.950 
521.519 
520. 
518. 6% 


517.16 
515.685 
514. 
512. 
511. 
509.67 
508. 1: 
506. 5S 
505. 
5038. 
501. { 
500. ¢ 


498.7 


TABLE VII.—LONG CHORDS. 


— 
/ Lone CHorDs. 
Degree | | 
| of 
Curve, s 9 | 10 11 12 
; Stations.| Stations. Stations. | Stations.| Stations. | Stations. 
. | | | 
10° 10 | 656.728 735.467 808 .426 875 .025 934.741 987.105 
: 20 | 655.320 733.887 805.495 371.058 | 929.542 980.47% 
30 | 653.895 731.277 802.524 867.038 924.276 973.760 
40 | 652.450 729.137 799.512 862.963 918.943 966. 967 } 
50 650.983 726. 967 796.458 858 . 836 913.544 960.0938 i 
: 11 649.496 724.767 793.364 854.656 908 . 080 953.141 | 
| 10 | 647.989 722.537 790 . 280 850.425 902.550 946.112 Wt 
| fs 20| 646.460 720.278 787.056 846.140 896.957 939.007 Mi 
80 | 644.911 717.990 783.843 841.808 891 .303 931.828 fi 
40 | 643.342 715.672 780.590 837.424 885.586 924.575 i} 
: 50 | 641.752 713.825 777.298 832.990 879.807 917.250 i 
| 12 640.142 710.950 773.968 828 .507 73.968 909 .854. Hl 
| 10 | 638.512 708.546 770.600 823.974 868.070 902.389 i 
: 20 | 636.862 706.113 767.193 819.394 862.1138 894.855 
: 30 | 635.191 703 .653 763.749 814.766 856.099 887.254 | 
: 40 | 683.501 701.164 760.268 810.092 850.028 879.588 } 
50 | 631.792 698.647 756.749 805 .370 843.900 871.857 i 
: 13 630.062 696.103 753.194 800.602 837.718 864.063 i 
: 10 328.313 693 .531 749.603 795.790 831.482 856.208 | 
20 | 626.544 690 . 932 745.976 790 .932 825.192 848 .293 it 
: 380 | 624.756 688.306 742.313 786 .030 818.850 840.318 | Wa 
: 40 | 622.949 685 . 653 738.616 781.085 812.457 832.286 nai 
: 50 | 621.123 682 . 974 734 883 776.096 806.013 824.198 , 
14 619.27 680.268 731.116 771.066 799 .520 816.056 i 
10 | 617.413 677.535 727.815 765.993 792.979 807.860 i 
20 | 615.530 674.777 723.480 760.879 786.389 799 612 th 
30 | 613.628 671.993 719.612 (55.725 779 753 791.313 i 
40 | 611.708 669.183 (abnareel 750.531 (73.072 782.966 i 
59 | 609.769 666.348 LOBE 745.297 766 .345 774.571 1 
15 607 .812 663.488 707.811 740.024 159.575 766.130 i 
10 | 605.836 660.603 703.814 734.714 752.763 
2 603.842 657.693 699.785 729 366 745 .908 . 
30 | 601.831 654.758 695.725 723 . 982 739.014 } 
40 | -599.801 651.799 691 . 634 718.561 732.078 
50 | 597.753 648.817 687.512 713.105 725 . 104. | 
16 595.688 645.810 683 .362 707.614 718 .092 |} 
10 | 593.605 642.780 679.182 702.088 711.043 1 
20 | 591.505 639.727 674.973 696.529 703.959 | 
30 | 589.388 636 .650 370.735 690.938 | 
40 587.253 633 .550 666.469 685 .314 im 
50 585.101 630.428 662.175 679.659 
17 582.933 627.283 657.854 73.972 
10 | 580.747 624.117 653.506 668 .256 ; 
20} 578.545 620.928 649.131 662.510 
30 | 576.326 617.717 644.730 656 735 
40 574.091 614.485 640.304 650.9338 
50 | 571.839 611.232 635.852 645.103 | 
18 569.571 607.958 631.375 | 639.245 
10 | 567.287 604.664 626.874 
20 | 564.988 601.3849 622.349 
30 | 562.673 598.013 617.801 
40 | 560.342 594.658 613.229 
50 | 557.996 591.283 608 . 635 
19 555 . 634 587.888 604.018 
10-4558. 257 | 584.475 599 .37$ 
20 550.864 581.042 594.720 (1 
30 548.457 577.591 590.039 
40 546 . 035 574.121 585.339 
50 543.599 570.634 580.618 | 
541.147 567.128 75.8% j : : 


Actual 
Are, 
One 


Station. 


100.562 
100.617 
100.675 
100.735 
100.798 
100.8638 
100.931 
101.002 
101.075 
101.152 


TABLE VII.—LONG CHORDS. 


Lone CHORDS. 


2 
Stations, 


196.651 
196.825 
195.985 
195.630 
195.259 
194.874 
194.474 
194,059 
193.630 
193, 185 


3 
Stations. 


286.716 
285.487 
284.101 
282.709 
281 .262 

79.759 
278.201 
276.589 
274, 924 
278.205 


pe RR RT WS 


4 
Stations. 


367.179 
364.060 
360.810 
357.483 
353 . 930 
350.303 
346 .555 
342,688 
338 .'704 
334.607 


5 
Stations. 


435.345 
429 305 
423.033 
416.5385 
409.819 
402.891 
395 . 758 
388 .428 
380.908 
373.205 


6 
Stations. 


488 . 931 
478.705 
468.270 
457 .433 
446.280 
434.827 
423 .092 
411.092 
398.846 
386.370 


— 


fora 
ry 
Fat 
=< 
Ps 
— 
fo 
eS 
© 
me 
— 
(exn) 
jam) 
jn 
i 
— 
— 
— 
me 
bol 
a) 
=< 
es 


TABLE VII.—MIDDLE ORDINATES. 


oe 
| 
YEE | 1 | 2 8 
Curve, | Station. | Stations. | Stations. 
.036 .145 232 
078 291 654 
.109 .436 .982 
145 .582 1.309 
.182 127 1.636 
218 .873 1.963 
255 1.018 2.291 
291 1.164 2.618 
327 1.309 2.945 
.364 1.454 | «827 
Tot e400 1.600 3.599 
.436 1.745 3.926 
Gate oc Sea ee ey 4.253 
| 509 «| 2.086 4.580 
545 2.181 4.907 
.582 2.827 5.234 
.618 2.472 5.561 
654 | 2.618 5.888 
694 2.763 6.215 | 
(27 2.908 6.542 
163 3.054 6.868 
.800 3.199 | 7.195 
.836 3.345 7.522 
.872 3.490 7.848 
|  .909 3. 635 8.175 
te. 945 3.781 8.501 
.982 3.926 8.828 
| 1,018 4.071 9.154 
| 1.054 4.217 9.480 
1.091 4.362 9.807 
1.127 4.507 10.188 
1.164 4.653 10.459 
1.200 4.798 10.785 
1.287 4.943 yb es ot Peas 
1.273 5.088 11.486 | 
1.309 5.284 11.762 
| 1.846 5.379 12.088 
1.382 5.524 12.413 | 
1.418 5.669 12.739 
1.455 5.814 13.064 
1.491 5.960 13.389 
1.528 6.105 13.715 
1.564 6.250 14.040 
1.600 6.395 14.365 
1.637 6.540 | 14.689 
1.673 6.685 | 15.014 
1.710 6.831 | 15.339 
1.746 6.976 15.663 | 
| 1.782 7.12 15.988 | 
1.819 7.266 16.312 
1.855 7.411 16.636 
1.892 7.556 16.960 
1.928 7.701 | 17.284 
| 1.965 7.846 17.608 | 
| 2.001 7.991 | 17.982 
2.037 8.136 18.255 
2.074 8.281 18.578 
2.110 8.426 18.902 
2.147 8.57 19.225 
2.183 19.548 


| 


es) 
Nl 9 
— 
oS 


4 5 6 
Stations. | Stations. |Stations. 
582 .909 1.309 
1..164 1.818 2.618 
1.745 2.727 | 3.926 
2.327 8.686 | 5.285 
2.909 4.545 | 6.544 
3.490 5.4538 | 7,852 
4.072 6.362 9.160 
4.654 7 270 10.468 
5.235 8.179 11.775 
5.816 9.087 13.082 
6.398 9.994 14.389 
6.979 10.902 | 15.694 
7.560 11.809 | 17.000 
8.141 12.716 | 18.304 
8.722 13.623 | 19.608 
9.303 14.529 | 20.912 
9.883 15.485 | 22.214 
10.464 16.341 | 23.516 
11.044 17.246 24 817 
11.624 18.151 | 26.117 
12.204 19.055 | 27.416 
2.784 19.959 | 28.714 
13.363 20.863 30.012 
13.943 21.766 31.308 
14.522 22.668 32.603 
15.101 23 57 33.896 
15.680 24.471 35.189 
16.258 25.372 36.480 
16.837 26.272 | 37.770 
17.415 27.171 | 39.059 
7992 28.070 | 40.346 
18.570 28.968 | 41.631 
19.147 29.866 | 42.916 
19.724 80.762 | 44.198 
20.301 31.658 | 45.479 
20.877 82.553 46.759 
21.453 33.448 | 48.037 
22.029 84.341 | 49.315 
22.604 35.234 | 50.587 
23.17 36.126 51.860 
23.754 7.017 | 58.180 
24.328 37.907 | 54.399 
24.902 38.796 55.666 
25 .476 39.684 56.931 
26.049 40.57 58.193 
26.622 41.458 59.454 
27.195 42.343 60.712 
27 767 a3 227 61.969 
28 338 44.110 63 . 228 
28.910 44.992 64.475 
29.481 45.873 65.72 
30.051 46.753 66.972 
30.621 47 632 68.216 | 
31.190 48.510 | 69.459 
31.759 49.386 70.699 
32.328 50.261 71.936 
32.896 51.135 "3 171 
33.464 52.008 74.403 
34.031 52.880 75.632 
34.597 53.750 76.859 


298 


Degree 
of 
Curve. 


O° 10’ 
20 
30 
40 
50 


10 
20 
30 
40 
50 


10 
20 


TABLE VUI.—MIDDLE ORDINATES. 


Tn el a 


7 


Stations. 


72.037 


Data otot= 


YO WF OUT 2 
or 00 
= 
oS 


® 
~) 

poh 

CO 


82.2 


12 


Stations. 


103.675 
108.747 
113.808 
118.841 


123.862 
128. 864 
133.847 
138.810 
143.753 
148.674 
153.572 
158.448 
163.300 
168.128 
172.931 
177.708 
182.459 
187.182 
191.878 
196.545 
201.183 
205.792 
210.370 
214.916 
219.431 
223.914 
228 .363 
232.7 

237.160 
241.507 
245.818 
250.093 
254.331 
258.531 
262.694 
266.818 
270.904 
274.949 
278.955 
282.919 
286.843 


11 


12 


13 


14 


15 


16 


17 


18 


19 


Degree 
of 
Curve. 


| 


Station. 


4.045 


cw Co Ww rw) 
© 
CO 


2 
Stations. 


3 6 
Stations. | Stations. | Stations. | Stations. 
19.870 35.164. 54.619 78.083 
20.193 35.729 55.486 79.805 | 
20.516 36.294. 56.3538 80.523 
20.888 36.859 57.218 81.739 
21.160 7.423 58.081 82.951 
21.483 37.986 | 58.943 84.161 
21.804 88.549 | 59.804 85.368 
22.126 39.111 | 60.663 86.571 
22.448 39.67% 61.521 87.705 
22.769 40 .284 62.377 88 . 969 
23.090 40.795 63 232 90.164 
23.412 41.355 | 64.085 91.3855 
23.782 41.914 64.9387 92.542 
24.053 42.473 65.787 93.727 
24.374 43.031 66. 656 94.908 
24.694 43 588 67.482 96 .086 
25.014 44.145 | 68.32 7.260 
25 .3834 44.701 69.17 | 98.431 
25.654 45.256 | 70.018 | 99.598 
25.974 45 .811 70.854 | 100.762 
26.298 46 .365 (12092, = 101Ge2 
26.612 46.919 72 .529 103.079 
26.931 47 472 73.364 104.232 
27.250 48 .024 74.197 105.381 
27.569 48 .575 75 .029 106.527 
27.887 49.126 75 . 859 107.669 
28 . 206 49.676 76.687 108 . 807 
28.524 50.225 77.518 109.941 
28.841 50.7% 78 3837 111.071 
29.159 51.32 79.159 112.197 
29.476 51.868 79.979 113.319 
29.794 52.414 80.798 114.488 
30.111 52.959 81.614 115.552 
30.427 53.504 82.429 116. 662 
30.744 54.048 83 . 241 117.768 
31.060 54.591 84.052 118.870 
31.37 55.133 84.861 119.967 
31.692 55.675 85 . 667 121.061 
82.008 56.215 86.471 122.15 
82.3823 56.755 8.274 128 .23 
32.638 57.294 88 . 074. 124.315 
32.953 57.882 88.872 25.891 
33 . 267 58 .369 89. 668 126.463 
33.582 58. 906 90 . 462 127.680 
33.896 59.441 91.254 128.593 
34.210 59.976 92.043 129.651 
84.523 60.510 2.83 180.704 
34.837 61.042 93.616 131.758 
85.150 61.57 94 398 132.797 
35.463 62.106 ae Yd 133 .&37 
35.775 62.686 95 957 134.872 
36.088 63.165 96.783 135 . 902 
36.400 63.693 97 .506 186 .$28 
36.712 64.221 98.278 137.848 
37.023 64.747 99.047 188.964 
37.3084 65.273 99.813 139.975 
87.645 65.797 100.577 140.981 
37.956 66.321 101.389 141 . 982 
38 . 266 66.843 102.098 142.978 
38.576 67.365 102.855 143.969 


TABLE VIII.—MIDDLE ORDINATES. 


aoe | 


TABLE IX.—LINEAR DEFLECTION TAPLE. 


Q ( 
| aoe | 900. | 1000. 
} / | | 
| sy | 0.87! 1.75] 2.62| 8.49] 4.386) 5.24) 6.11] 6.98) 7.85) 8.73 
ph cape 1.75| 8.49] 5.24} 6.98] 8.73) 10.47] 12.22] 18.96) 15.71) 17,45 | 
3) | 2.62| 5.24/ 7.85] 10.47] 13.09] 15.71) 18.38] 20.94) 23.56] 26.18 i 
2 3.49) 6.98) 10.47] 13.96] 17.45] 20.94] 24.43) 27.92) 31.41) 34.90 | 
39 | 4.36! 8.72| 13.09/ 17.45] 21.81] 26.18) 30.54| 34.90) 39.27) 43.63 i 
ee eo Ag 5.24| 10.47] 15.71| 20.94] 26.18] 31.41} 36.65/ 41.88) 47.12| 52.85 a 
30 | 6.11| 12.22] 18.32] 24.43] 30.54] 86.65) 42.75) 48.86) 54.97) 61.08 Hi 
4 6.98] 13.96] 20.94) 27.92] 34.90} 41.88) 48.86] 55.84) 66.82) 69.80 I 
30 | 7.85) 15.70) 23.56] 31.41) 39.26] 47.11] 54.96] 62.82) 70.67) 78.52 1A 
rl: ob 8.73) 17.45) 26.17) 84.80} 43.62! 52.34] 61.07) 69.79] 78.51) 87.24 | 
30 | 9.60! 19.19! 28.79] 38.33] 47.98] 57.57] 67.17| 76.76] 86.36] 95.96 Hi 
6 10.47| 20.93' 31.40] 41.87] 52.34] 62.80] 73.27| 83.74] 94.20) 104.67 1H 
i 30 | 11.34] 22.68] 34.02} 45.35] 56.67] 68.03} 79.37) 90.71] 102.05) 113.39 | 
7 12.21) 24.42! 36.63] 48.84] 61.05] 73.26) 85.47| 97.68] 109.89} 122.10 i 
| 30 | 13.08) 26.16) 39.24) 52.32] 65.40] 78.48] 91.56| 104.64] 117.73} 130.81 i 
8 3.95| 27.90) 41.85) 55.80} 69.76] 83.71) 97.66 | 111.61} 125.56) 139.51 i 
| 30 | 14.82| 29.64) 44.47) 59.29] 74.11] 88.93) 103.75 | 118.57| 133.40 | 148.22 il 
| 9 | 15.69} 31.38) 47.08) 62.77} 78.46] 94.15} 109.84 125.53} 141.23) 156.92 i 
| 30 | 16.56) 33.12) 49.68) 66.25} 82.81) 99.37 | 115.93) 182.49] 149.05) 165.62 i 
| 10. =|: 17.43} 34.86) 52.29) 69.72] 87.16] 104.59 | 122.02) 189.45] 156.88 | 174.31 ! 
30 | 18.30] 36.60] 54.90) 73.20] 91.50] 109.80) 128.10 | 146.40) 164.70) 183.00 |} 
| 11 19.17|. 38.34] 57.51) 76.68] 95.85 | 115.01 | 184.18 | 153.35] 172.52] 191.69 i 
| 30 | 20.04) 40.08) 60.11] 80.15] 100.19] 120.28 | 140.26 | 160.30] 180.34) 200.38 | 
12 =| 20.91} 41.81] 62.72} 83.62] 104.53 | 125.43 | 146.34 | 167.25/ 188.15 | 209.06 i 
30 | 21.77! 43.55] 65.32) 87.09! 108.87 | 130.64] 152.41 | 174.19] 195.96 | 217.73 I 
13 22.64) 45.23) 67.92) 90.56] 113.20] 185.84] 158.48 | 181.13) 203.77 | 226.41 H 
39 | 23.51| 47.01] 70.52| 94.03] 117.54 | 141.04 | 164.55 | 188.06] 211.57 | 235.07 i 
14. =| 24.387} 48.75] 73.12} 97.50] 121.87 | 146.24 | 170.62} 194.99) 219.36 | 243.74 I 
30,| 25.24) 50.48) 75.72) 100.96) 126 201 151.44 | 176.68 | 201.92] 227.16 | 252.40 i 
| 15 | 26.11] 52.21) 78.32) 104.42] 130.53 | 156.63 182.74 | 208.84) 234.95 | 261.05 } 
30 | 26.97) 53.94] 80.91) 107.88} 134.85 | 161.82 | 188.79 | 215 76) 242.73) 269.70 
| 16 | 27.83] 55.67) 83.50) 111.34] 139.17] 167.01 | 194.84 | 222.68) 250.51 | 278.35 
30 | 28.70| 57.40} 86.10) 114.79| 143.49 | 172.19 | 200.89 | 229.£9 | 258.29 | 286.99 
17 29.56) 59.12] 88.69] 118.25 | 147.81 | 177.37 | 206. 93 | 236 .50| 266. 06| 295.62 
30 | 30.42) 60.85] 91.27/ 121.70| 152.12 | 182.55 | 212.97 | 243.40) 273.82 | 304.25 1] 
18 31.29} 62.57] 93.86) 125.15] 156.43 | 187.72 | 219.01 | 250.30] 281.58 | 312.87 
30 | 32.15] 64.30] 96.45] 128.59] 160.74 | 192.89 | 225.04 | 257.19 | 289.34 | 321.49 
19 33.01) 66.02) 99.03) 132.04] 165.05 | 198.06 | 231.07 | 264.08 | 297.08 | 830.09 | 
30 | 33.87) 67.74 101.61] 135.48] 169.35 | 203.22 | 237.09 | 270.96 | 304.83 | 338.7 
20 34.73 69.46 1104.19 138.92] 173.65 | 208.38 | 243.11] 277.84 | 812.57 | 347.30 al 
30 | 35.59) 71.18 /106.77| 142.35] 177.94 | 213.53 | 249,12 | 284.71 | 320.30 | 355.89 | 
21 |: 36.45) 72.89 109.34) 145. 79| 182.24) 218.68 | 255.13 | 291.58] 828.02 | 364.47 \ 
30 | 37.30] 74.61 /111.91] 149.22] 186.52 | 223.83 | 261.18 | 298.44 ) 835.74 | 873.05 |} 
22 38.16] 76.32 |114.49| 152.65| 190.81 | 228.97 | 267.13 | 305.29 | 343. 46 | 381 62 | 
30 | 39.02] 78.04/117.05| 156.07] 195.09 | 234.11 | 273.13 | 312.14|351.16 | 390.18 . 
93 39.87| 79.75 |119.62) 159.49] 199.37 | 239.24 | 279.12 | 818.99 | 858.86 | 398.74 | 
30 | 40.73) 81.46 122.19) 162.91} 203. 64 | 244.87 | 285.10 | 825.83 | 866.56, 407.28 
94 41.53) 83.16 124.75) 166.33] 207.91 | 249.49 | 291 .08 | 832.66 | 874.24 | 415.82 
30 | 42.44) 84.87 |127.31 169.74| 212.18 | 254.61 | 297.05 | 839.48) 881.92] 424.36 [| 
25 43.29 86.53 |129.86) 173.15] 216. 44 | 259.73 | 803 .02| 864.80 | 889.59 | 432.88 | 
| 80 | 44.14) 88.23/132.42) 176.56] 220.70 | 264.84 | 808.98 | 853. 12|897.26 | 441.39 
26 44.99| 89.98 134.97 179.96) 224.95 269.94 814.93 |359.92| 404.91 | 449.90 
30 | 45.84! 91.68 /137.52) 183.36] 229.20 | 275.04 | 820.88 | 866.72 | 412.56 | 458.40 
27 46.69} 93.38 /140.07) 186.76] 233.45 | 280.14 | 826.82 | 873.51| 420.20 | 466.89 
30 | 47.541 95.07 |142.61| 190.15) 237.69 285.22 | 332.76) 380.30 | 427.83 | 475.37 . 
28 48.38| 96.77 /145.15/ 193.54| 241.92 | 290.81 | 838. 69| 887.08 | 435.46 | 483.84 
30 | 49.23| 98.46 /147.69 | 196.92) 246.15 295.33 | 844.62) 893.85 | 443.08 492.31 . 
29 50.08 1100.15 150.23, 200.80] 250.38 800.46 | 350.53] 400.16 | 450.68 | 500.7 
30 | 50.92/101 .84/152.76 203.68) 254.60. 305.52 | 356.44 | 407.36 | 458.28 | 509.20 
30 51.76|103.53 155 .29| 207 .06| 258.82. 310.59 | 8362.35] 414.11 1465.87 | 517.64 


TABLE X.—COEFFICIENTS FOR VALVOID ARCS. 


Oy 


Z 
I.—RaTIO OF u = —: 
A 


—— 


3337 | 38336 


3343 
8339 


| 3366 | .8364 
. 8358 | .8356 
. 3352 | 3350 
- 3348 | 3346 

3341 

. 3337 | .3335 

.3333 | 3331 


L 10° | 20° | 30° | 40° | 50° | 60° | «0° | 80° | 90° | 100° | 110° | 120° 
| | posts. | | 

300 | .8518 | .3516! .3514|.3510| 3506) .3500 .8493} 3485 | 3476 | 3466 | .3455| .3444 

400 | .3437 | 3436 | .3433 | .3430| .3426] .3421 | .3415) 8408} .3399 | .3390| .3380) .3368 

500 | .3400 | .3398 | .3396 | .3393,] 8389) . 3383 | .3379] :3372| 3364 | .3856 8845} 3335 

600 | .3879 | .38878 | .3376 |.3373 | 8369] .38365 | .38359) 8353] .8345 | .38337 | .8827| .3317 


8361 
8359 
8348 
3344 
.8339 


3357 
83849 
3844 
8340 
.3336 
3831 
3328 


8353 | 3347) . 
3345 | .38840 

8340 | .3384! . 
.8336 | 3331) . 
.8931 | 3326} . 
8827 | 8822) . 


. b024 | 8319 


3334 
83826 
3821 


8326 
8318 
.8318 
.3317 | .38310 
.3313 | .8805 
.3309 | .3301 
. 3306 | 38298 


.3316 | 
. 3809 
. 3304 
83801 
8296 
8292 


. 38289 


8306 | 
8299 | 
8294 
8291 
.3286 
8283 
3280 


co 
2 
Ww 
Co 


I.—RATIO oF 


L 


10° 


tide 
| 


30° 


90° | 100° | 110° | 120° 


80° 


400 
500 
600 
700 
800 
900 
1000 
1200 
1500 
2009 


300 


7706 | 
7611 |. 


z 
‘ 

F 

|. (545 | 7506 | 7452 | .7384 | .'7 
F 

a 

‘ 


. 1683 | 7648 | .7588 | .'7518) . 
7588 | .7549 | .'7495 | 7425 | .7 
7522! 7483 | 7430] 72 
| 7508 |. 7 7 

22 | 7499 | .'7461 | 7407) . 7 
| 7492! 7454 | .7401 | . 7338 

2} 7489 | 7450 | 7397 | .7329 | .'7% 
05 | .'7483 | .7444 | 7391 |. 732 
W501 | 17478 | 7440 | 7387 | 731! 
bak Me be 7436 piaga ta 


| 7218 | 7090 | .6949 | .6795 
7130 | 7004 | .6865 | .6714 | 
2). 7091 | .6966 | 6828) .6678 | 
7070 | .6946 | .6808] . 6659 | 
.7057 | .6933 | .6797'| .6648 
.7049 | 6926 | .6789) 6640 | 
7044 | 6920 | 6784 | 6635 | 
(040.6917 6780) . 6632 | 
5| 7035 | 6912) .6775 | . 6627 | 
|. 031 | .6908 | .67'72 | . 6624 | 
028 |-6904) .6760) .6621 


F 2: a | | 


. 6630 
6551 
.6516 
6498 
.6487 
.6480 
6475 
6472 
.6468 
6464 
.6461 


469 | .7416| . 7 


Tii.—Ratrio'or §, = —— = 
A’—A 
TO A CHANGE OF ONE DEGREE IN THE 


l 
— = LENGTH OF VALVOID ARC CORRESPONDING 


ANGLE A. 


L 


re 
f=) 
fe} 


100° 


300 
400 
500 
600 
700 
800 
900 
1000 
1100 
120) 
1300 
1400 
1500 
1600 
1700 


1800 
1900 


. 2000 


CO OO 3 Od CI OT PB GY 0 


SOOO 
.46 |10. 
.69/11.: 
.21)12. 
.08 13. 
9513-5 
182 |14.7 
.69 |15.65 
sO 10605 ; 
4417.39.17. 


62 | 61) 

| 3.48 
36} 35 
23) 5 2 | 
10| 6 


rey 
es 
3D SD OT HR CO 09 


97) 6 
85| 7 .82| 
72| 8.69| 


=) 
a) 
© WH 38 OD SI O1 ® 09 09 i 
5 lo} 
| 
bss | > 
on 6 a 
a => 
o or 
j=) Co Go GS GoW 
mat ° Rie 0 ww co 
*; OO GIO 
IS? OT OU GY 0 ior) J 9 9-9 3 
oe B W WW WW 
wg Go CO HS HS OT Or 
CO OI mae ses 
co J J I 
“ S peer et et 
ee < Sas 2 OOH PA OUOT DD if : 
y ROH ReTORS DG 
e0) 
Oo 
a ° 
oO 
2} le) 
(<5) 
~ fo) 
el 
_ 
a 
toh Eilers 
(—p) 
a ° 
aa 
oO 2 
co 
Qo ° 
neg 
i 


ery 
co 


ror) 

O DFO DOr 09 
OMI ODOT ROO e 
oS 
oo 
COWDNIOUP KW 

~ | 


oie 
ROO OID wr 
CM) 
i) 
WOO WOIO DOB OCD 


11 |12.03)|11.93 }11.80|11.65 11.48 111.29 86| 10.61 
98 |12.89 }12.78 /12.64|12.48 12.30 112.10) 11.88/11.63! 11.37 
84 }13.75 |13.63 }13.49 |13.32|13.12 12.91} 12.67 12.41} 12.13 | 
71 |14.61)14.48 |14.33}14.15 13.94 18.71 | 13.46/13.18| 12.88 
57 15.47/15 88 |15.17|14.98 |14.76 14.52/14.25 113.96) 13.64 


).01 15.81 
3.86 16.65 


115.58 15.3: 
16.40 16. 


30 |17.19}17.04| 


302 


TRACK. 


TABLE XI.—TURNOUTS AND SWITCHES FROM 
§§ 180, 181, 182. 


A STRAIGHT 


GAUGE, 4 FEET i SyorEnS- = 4, iOS: 


THRow, 5 INCHES = 0.417: 


No Angle Dist. | Chord | | Switch | Radius | Log’thm.| Degree 
N. ole BF. | af. AD De log. r. | of Curve. 
4 | 14° 15/ 00" | 387.664; 387.8738) 11:209) - 150.656 2.177986 | 38° 45/5 Die 
| 414 |12 40 49| 42.372) 42.113 12.610, 190.674 | 2.280292 | 30 24 09 
: 5 11 2% 16 7.080 | 46.846| 14.012] 235.400 | 2.371806 | 24 31 36 
544 | 10 23 20 51.788 | -51.575 | 15.413 284.834 2.454592 | 20 18 13 

: Gy eed. 31.539 56.496 | 56.301 16.814 | 338.976 | 2.580169 | 16. 57 52 
| 6146 | 8 47 51 61.204) 61.024 18.215 | 397.826 2.599693 | 14 26 25 
ii 8 10 16 65.912 | 65.744 19.616 461.384 2.664063 | 12 26 34 
: 16 | 7 37 41 | 70.620) 70.464) 21.017 | 529.650 | 2.723989 | 10 50 02 
: 8 7 09°10 |. 75.828 | 75.181} 22.418 602.624 | 2.780046} 9 31 07 
) 8144 | 6 43 59 80.086 | 79.898} 28.820 680.3806 2.882704 8 25 47 
: 9 6 21 35 84.744 | 84.613] 25.221 762.696 2.882352 7 31 04 
: 916 | 6 01 32] 89.452) 89.328) 26.622 849.794 | 2.929814 | 6 44 46 
: 10 5 43 29 94.160} 94.048} 28.023 941 .600 2.973866 6 05 16 
: 104 | 5 27 09 98.868 | 98.756 29.424 10388 .114 3.016245 5 381 17 
11 5 12 18 | 108.576 | 103.469 30.825 1139 .336 3.056652 5 01 50 

: 11% | 4 58 45 | 108.284} 108.182 | 32.227 1245 ..266 8.095262 4 36 08 
: 12 4 46 19 | 112.992 | 112.894) 33.628] 1855.904 | 3.182229) 4 138 36 
: | z 

GAUGE, 3 Feet. Turow, 4 INCHES = == 0) 338, 

No Angle Dist. | Chord , Switch! Radius | Log’thm. | Degree 
n. | Ff. BF, 55 2 OL (a log. r. of Curve. 
4 | 14°15’ 00" 24 «| 23.815; 8 96.0 1.982271 | 62° 46/ 24” 

4t46 | 12 40 49 27 =| 26.835 | ) 121.5 2.084576 48 36. 04 

5 11 25 16 30 =| 29.851 10 150.0 2.176091 38 56 35 

516 | 10 23 20 3: 32.865 | 11 181.5 2.258877 | 31 58 55 

6 9 31 39 36 SOOO le 216.0 2.884454 26 46 07 

616 8 47 51 389 38.885 13 258.5 2.403978 22 45 04 

4 8 10 16 42 41 893 14 294.0 2.468347 19° 35. Of 

76 tot 4) 45 44.900 15 307.5 2 -ER8274 17 02) 21 

Teo! eOOE LT 48 47.906 16 384.0 2.584331 14 57 48 

86 | 6 43 59 51 50.912 17 433.5 2.686989 3 14 47 

9 6 21-35 54 53.917 18 486.0 2.65 6686 11 48 3% 

944 OxO1- 32 Sf 56.921 19 541.5 2.736 10 35 46 

10 5. 48 29 60 59.925 20 600.0 2. 9 383 38 

101, 5 27 09 3 62.929 21 661.5 2. 8 40 12 

11 5 12 18 66 65.932 22 726.0 2.860937 (ents 37 

114% 4 58-45 69 68.935 23 193.5 2.899547 % 13 32 

12 4 46 19 72 71.988 24 864.0 2.986514 6 &8 06 

ANGLE AND DisTANCE OF MIDDLE FROG, F"" 

| Gauge Gauge | Gauge | Gauge 

No.| No. | Angle | 4,84. 3. || No.| No. Angle | 4,8%.| _ 3. 

nd. nv" ys Dist. Dist. || 7, qu". he Dist Dist. 

GE. \aie*s 4 aF'' ak; 

4 | 2.817} 20° 07’ 36"| 26.786 | 17.037 || 8 5.651 | 10° 06! 44"| 53.817 | 33.97 

416; 3.172) 17 54 52) 30.054 | 19.151 |} 814) 6.005| 9 31 08) 56.643 | 86.094 

5 8.021| 16 08° 19) 33.374) 21-266 9 6.359 | 8 59 80] 59.969 | 88.213 

5 6) 3.881; 14 40 58] 36.695 | 23.383 || 9146) 6.713| 8 81 10] 63:296 | 40.333 

G6 | 4.235) 138 27°57 | 40.018 | 25.500 |} 10 | 7.067 | 8 05 40] 66.623 | 42.45: 

614| 4.589) 12 26 07} 43.342.| 27.618 || 1014) 7.420) 7 42 35| 69.950 | 44.57 
% | 4.943) 11 83-04] 46.666 | 29.736 i Gee a ae 7 21 36) 73.277 | 46.693 
746) 5.297| 10 47 02) 49.991 | 31.855 || 1116) 8.128] V 02 26] 76.605 | 48.813 

8 | 5.651) 10 06 44/1 58.317 | 33 974 2 | 8.482 | 6 44 51} 79.982 | 50.984 


ape ee — 
2 


TABLE XII.—MIDDLE ORDINATES FOR CURVING RAILS. 


LENGTH OF RAIL-CHORD. 
D : a D 
a2 | so { 28 | 26 | 24 | 22.| 20 | 48 | 16 | 14 | 12 | 10 
1°} .022| .020| .017} .015| .013| .011| .009 .007 |.006 |.004|.003 | .002| 1° 
2} .045 | .039] .034) .030| .025 .021 | .017| .014 |.011 | .009 | .006 | 004 2 
3 | .067| .059} .051) .044} 038) .032| .026} .021 |.017 |.013 |.009 | .007| 3 
4 | .089| .079| .068| .059| .050 .042 | .035 | .028 |.022 |.017 |.013 | .009| 4 
5 | .112} .098| .086| .074| .063| .053) .044) .035 |.028 | .021 /.016 | 011] 5 
6 | .134| .118} .103 | .088| .075 | .063| .052) .042 | .034 | .026 | .019 | 013] 6 
7 | -156| .137| .120 | .103| .088 | .074 | .061 | .049 | .039 |.030 |.022 | 015) 7 
8 | .179| 157] .187 |) .118) .100] .084| .070] .057 | 045 | .034 |.025 | .017| 8 
9 | 201} .177) .154] .133) .113] .095 | .078| .064 |.050 |.038 |.028 | .020] 9 
10 | .223| .196 | 171 | .147 | .126] 105) .087| .071 |.056 | .043 | 031 | .022 | 10 
11 | .245} .216| .188| .162] .138 | .116 | .096 | .078 |.061 |.047 | 035 | 024 |. 11 
12 | .268) .235| .205| .177| .151| .127| .105 | .085 |.067 |.051:| 038 | .026 | 12 
14 | B12} .274| .238 | .206) .175 | .147 | .122 | .099 |.078 |.(60 |.044 | .030| 14 
16 | .856| .313 | .273 | .235 | .200| .168 | .139| .113 |.089 |.068 | .050 | .035 | 16 
18 | .400} .352| .307 | .264) .225 | .189 | .156| .127 |.100 | .077 |.056 | .039 | 18 
20 | 445) .3891 | .3840) .293| .250 | .210 | .174| .141 |.111 |.085 |.063 | .043 | 20 
24] 531) .467| .407) .851 | .299 | .251 | .207| .168 |.133 |.102 | .075 | .052 | 24 
28 | 618) .543| .473| .408 | .347| .292 | .241 | .195 |.154 |.118 |.087 | .060| 28 
2 | .705| .619| .539| .465 | .396 | .833 | .275 | .223 |.176 |.135 |.099 | .069 | 32 
36 | .791| .696| .606 | .522! .445 | .873 | .309| .250|.197 |.151 |.111 | .077 | 36 
40 | .878| .772 | .672 | .579 | .493 | .414 | .3842 | .277 |.219 |.168 |.123 | .086 | 40 
45 | .983| .863| .752| .648) .552| .463 | .883 | .305 |.245 |.188 |.137 | .096 | 45 
50 (1.087 | .955| .831 | .716) .610| .512 | .423] .343 |.271 |.207 |.152 | .106 | 50 
TABLE XUI.—DIFFERENCE IN ELEVATION OF RAILS ON 
CURVES, §201. . 
VELOCITY IN MILES PER Hour. 
D D 
10; 15 | 20 | 2 | 30 | 3 40 45 50 60 

1 | .006 | .013 | .023 | .036 | .051 | .070| .091 | .116| .143| .206| 1 
2| .011 | .026 | .046 | .O71 | .103 | .140 | .183 | .281 | .285 | .410] 2 
3 | .017 | .039 | .069 | .107 | 154.) .210) .274] 846] .427) 612] 3 
4 | .023 | .051 | .091 | .143 | .206 | .280| .865] .461 |] .568| 811] 4 
5 | .029 | .064 | .114 | .179 | .257 | .849 | .455 | .574 | .707 | 1.006 | 5 
6 | .034 | .077 | .137 | .214 | .808 | .418 | .545 | .687 | .844| 1.196 | 6 
7 | .040 | .090 | .160 | .250 | .859 | .487 | 1634 | .798 | .97 

8 | .046 | :103 | .183 | .285 | .410 | .556 | .723 | .908 | 1.112 

9 | C51 | .116 | .206 | .820 | .460 | .624 ; .811 | 1.017 

10 | .057 | .129 | .228 | .856 | .511 | .692 | .898 | 1.124 

11 | .063 | .142 | .251 | .891 | .561 | .760 | .984 

12 | .069 | .154 | .274 | .427 | .611 | .826 | 1.069 

30 | . 818 ‘11 | .959 
: /1.088 


io) 
cal 
— 
Zi 
re 
— 
<< 
=} 
i 
om) 
= 
<— 
iva) 
i=) 
— 
— 
os 
T 
— 
amma 
bet 
<= 
<e 
2Q 
=< 
& 


TABLE XIV.—GRADES AND GRADE ANGLES. 


| | 

ae | Feet | Feet 
per | Feet per | Inclina-|| per | Feet per | Inclina- || per | Feet per |Inclin- 
Sta-| Mile. tion. || Sta-| Mile. tion. || Sta-| Mile. | ation. 

tion tion tion 
lo se So e/ ” } (oe URL 
01 528 21 .51 | 26.928 17 32 1.01 | 53.328 34 48 
2 1.056 41 .52 | 27.456 17 53 1.02 | 53.856 35 04 
03 1.584 1 02 .538 | 27.984 18 18 || 1.08 | 54.384 35 24 
04 2.112 1 23 .54 | 28.512 18 34 || 1.04 | 54.912 35 45 
.05 2.640 1 43 .55 | 29.040 18 54 || 1.05 | 55.440 36 05 
.06 3.168 2 04 56 | 29.568 19 15 || 1.06 | 55.968 36 26 
07 3.696 2 24 57 | 30.096 19 86 || 1.07 | 56.496 36 47 
.08 4.224 2 45 58 | 30.624 19 56 |' 1.08 | 57.024 37 08 
09 4.752 3 06 59 | 31.152 20 17 1.09 | 57.552 87 28 
10 5.280 3 26 60 | 31.680 20 88 || 1.10] 58.080 37°49 
11 5.808 3 47 61 | 382.208 20 58 1.11 | 58.608 38 09 
12 6.336 4 08 62 | 32.736 21.19 || 1.12 | 59.136 388 30 
13 6.864 4 28 63 | 33.264 21 389 |) 1.138 | 59.664 38 51 
14 7.392 4 49 64 | 33.792 22 00 || 1.14 | 60.192 39 11 
15 7.920 5 09 65 | 34.320 22:21) || 1:15") °60:720 39 32 
.16 8.448 5 30 66 | 34.848 22 41 1.16 | 61.248 39 53 
mls 8.97 Ns Se BSE 25°02 SHAT bie 1G 40 13 
.18 9.504 6 11 68 | 35.904 23 23 || 1.18 | 62.304 40 34 
.19 | 10.032 6 32 69 | 386.4382 23 43 || 1.19 | 62.882 40 54 
20 | 10.560 6 53 70 | 36.960 24 04 1.20 | 63.360 41 15 
21 | 11.088 7 13 71 | 387.488 24 24 1.21 | 68.888 41 35 
22 | 11.616 7 34 72 | 388,016 24 45 || 1.22] 64.416 41 56 
23 | 12.144 7 54 7 38.544 25 06 1.23 | 64.944 42 17 
24 12.672 8 15 74 39.072 25 26 1.24 65.472 42 38 
25 | 13.200 8 36 7 39.600 25 47 1.25 | 66.000 42 58 
26 | 13.728 8 56 7 40.128 26 08 1.26 | 66.528 43 19 
R27 | 14.256 9 17 77 | 40.656 26 28 1.27 7.056 43 39 
28 | 14.784 9 38 7 41.184 26 49 1.28 7.584 44 00 
29) 15.812 9 58 7 41.712 27 09 1.29 | 68.112 44 21 
30 | 15.840 1019 || .80 | 42.240 7 30 1.30 | 68.640 44 41 
31 | 16.368 10 39 .81 | 42.768 27.51 || 1.81 | 69.168 45 02 
82 | 16.896 11 00 .82 | 43.296 28 11 || 1.82) 69.696 45 23 
33 7.424 ib hepa .83 | 43.824 28 32 1.83 | 70.224 45 43 
34 | 17.952 11 41 .84 | 44.352 28 53 1.34 | 70.752 46 04 
85 | 18.480 12 02 .85 | 44.880 29 13 1.85 |. 712280 | 4624 
36 | 19.008 12 23 .86 | 45.408 29 34 || 1.386] 71.808 46 45 
7 | 19.536 12 43 .87 | 45.936 20 54 iad 87% leiecsae 47 06 
38 | 20.064 13 04 88 | 46.464 80 15 || 1.38]. 72.864 7 26 
39 20.592 13 24 89 46.992 30 36 1.39 73.392 7 47 
40 | 21.120 13 45 90 7.520 30 57 || 1.40 | 73.920 48 08 
41 | 21.648 14 06 .91 | 48.048 31 17 1.41 | 74.448 48 28 
2 22.176 14 26 .92 48.576 81 38 || 1.42 74.976 48 49 
43 22.704 14 47 93 49.104 31 58 1.43 75.504 49 09 
44 | 23.232 15 08 94 49 .632 32 19 || 1.44 76.032 49 30 
45 | 23.760 | 15 28 -95 | -50.160 382 39 || 1.45 | 76.560 49 51 
46 | 24.288 15 49 .96 50.688 33. 00 1.46 77.088 50 11 
47 | 24.816 16 09 977) 51, 216 33 21 1.47 | 77.616 50 382 
48 | 25.344 16 30 .98 | 51.744 83 41 1.48 | 78.144 50 52 
49 | 25.87 16 51 .99 | 52.272 34 02 1.49 | 78.672 51 13 
50 | 26.400 uci a 1.00 | 52.800 34 23 || 1.50 | 79.200 51 34 


TABLE X1IV.—GRADES aND GRADE ANSLES, 


Feet per Inclina- || per | Feet per 


WNWNWWNKTWWWWew 


D9 aA 0 
Sete BAT 


wopee wapapaoraprw 
2g Oe a3 


OPAC 


Co 09 09 C8 
1) 
fan io) 


SOODDYIRIBMONM 
STMONSMAS WAS & 


woe 
on) 


frek fe pk peek ek pe ek pe pe ek ek ek ek ek ek ek ek pk pk pe ek eek fk pe 


CORRES RRR RR He C9 C2 CO CO 09 GO GH 09 2 


DD et ek et et ed et ek et ek ek ek dk ek ek ek ek ek ek ek tk et ek ek at eed. Pek Pek Ret et ek et Bk ek Ped Ped Rt 
Ne ee ae os : . pele Ey Ear aui, ae, ah -@ ° eae ee we eh ees ° aie ca eee es Neots ie Sap 
- m Ae We a SiG 3 5 sone a 2A 4 > he xy se, he 


Mile. 


| 


108.240 | 
110.880 

118.520 | 
116.160 
118.800 
121.440. | 
124.080 
126.720 | 


129.360 


7 
(3) 
NN 
for) 
(—) 


— 
=, 
or) 
[ey] 
rp) 
© 


et 
~3 

io 2) 
forge) 
[=Jer 
oom 


© 29 
OW 
oo =~2 

> o9 D> 
SS WS 
SS 


258 ..720 


264.000 


OVO OT OT OTT 
BOR WWE 
SSSSS6Sso 


WM WNWNWNWNWNWNWWWD WDHB HHH eee Yee ee Bee eee ee ee Oo 
a ere S rN?) SS Par Cm 


wa 


we 


2D 0O 


VW WW We 


we 


2, 


Feet per | 


Mile. 


269.280 | 
274.560 
279.840 | 
285.120 
290. 400 
295. 680 
300.960 
306.240 
311.520 
316.800 


322.080 
327.860 
352.640 
337.920 
843.200 | 
348 .4&0 | 
353.760 | 
359 .040 


364.820 | ¢ 


369.600 


374.80 | ¢ 


380.160 
8&5 .440 
360.720 
306 .C00 
401.280 
406 .5€0 
411.840 
417.120 
422.400 


306 


| 


gcc cco ce to CCV cmH CU CU CHE CUCU WW Y 


mb eww WMeRH He OOo 
MPWOMWO BDBMWOOUW MW Mor > 
rc 


OT OT OTe 


Or OV OT Or OF OF Ot OF OF O1 


TABLE XV.—FOR OBTAINING BAROMETRIC HEIGHTS IN FEED. 


Barom- Dur 
eter. 23 0 ' | Diir. per 
Traits 0.00 0.02 0.04 0.06 0.08 “002 fe 
ay (i 
ae aes : 
19°.0 16832 16860 16888 16915 16943 2.8 
a1 16970 16997 17025 17052 17080 2.8 
ae 17107 17134 17162 17189 7216 - 2 
33 17248 47270 17298 17325 17352 23% 
4 17379 17406 17483 17460 17487 2.7 
5) 17514 17540 17567 17594 17621 230 
.6 17648 T674 17701 17728 17755 22% 
=F 17781 17808 17834 17861 17887 enki 
8 17914 17940 17967 7993 18020 2.7 
9 18046 18072 18099 18125 18151 2.6 
20°.0 1817 18204 18230 18256 18282 2.6 
1 18308 18334 18360 18386 18413 2.6 
32 18438 18464 18490 18516 18542 2.6 
iS 18568 18594 18620 18645 18671 2.6 
\ 4 18697 18723 18748 1877 18799 236 
ati 5 18825 18851 18376 18902 18927 2.6 
Hit .6 18953 18978 19004 19029 19054 rates 
i 7 19080 19105 19130 19156 19181 2.5 
| 8 19206 19231 19256 19282 19307 2.5. 
i 9 19332 19357 19382 19407 19432 2.5 
21°.0 19457 19482 19507 195382 19557 Pia) 
a! 19582 19606 19631 19656 19681 2.5 
2 19706 19730 19755 19780 19804 2.5 
aa 19829 19854 19878 19903 19927 2.5 
4 19952 19976 20001 20025 20050 2.5 
35 20074. 20098 20123 20147 20172 250 
6 20196 20220 20244. 20269 20293 2.4 
are 20317 20341 20365 20389 20413 2.4 
8 20438 20462 20486 20510 20534. 2.4 
Hi 9 20558 20581 20605 20629 20653 2.4 
} | 222°0 20677 20701 20725 20748 20772 2.4 
tk 1 20795 20820 20843 20867 20891 2.4 
i 12 20914. 20938 20952 20985 21009 aA: 
iit 3 21032 21056 21079 21103 21126 2.4 
iy i 4 21150 21173 21196 21220 =| ~— 21243 2.3 
Wig 5 21266 21290 21313 21336)- ..| 2521859 2.3 
| 6 21383 21408 21429 21452 21475 Bae 
Ha ae 21498 21522 21545 21568 21591 2.3 
Wit 8 21614 21637 21660 21683 21706 Pe 
Palin| 9 21728 21751 2177: 21797 21820 es 
Hil 23°.0 21843 21866 21888 21911 21934. Te 
Ht 1 21957 21979 22002 22025 22047 2.3 
VHT 2 22070 92022 92115 22138 | 22160 2.3 
init io 22183 22205 22228 22250 22972 ue 
ij 4 22295 92317 22340 22362 22384 Dae 
Wi 5 22407 92429 22451 22474 22496 9.2 
i | 6 22518 22540 22562 22585 22607 Pe 
sai ai 22629 292651 22673 22695 22717 2.2 
8 22739 22761 22783 22805 99907 2.2 
i) 9 22849 92871 22898 92915 22937 2.2 
24°.0 22959 22981 23003 23024 23046 272 
1 23068 23090 23111 23133 23155 Psy 
Hit 2 23176 23198 93220 93941 23263 9.2 
iy il 3 23285 23306 93328 23349 93371 2.2 
4 23392 93414 23435 23457 23478 2.2 
. 5 23500 23521 23542 23564 23585 2.1 
i 6 23606 23628 23649 23670 23692 24 
i} “fe 23713 23734 93755 23776 | 23798 Ost 
WA | | 8 23819 23840 23861 23882 23903 2.1 
9 23924 23945 23966 23987 24008 ot 


307 


TABLE XV. 


—FOR OBTAINING BAROMETRIC HEIGHTS IN FEET. 


Barom- 
eter. 
Inches 
nai (Ie 


25° 


26°. 


. 
[ry 


27°. 


“rs 


~ 


28°. 


29°. 


30°. 


las) 


pean) 


0.00 


0.02 


0.04 


0.06 


0.08 


WOIRAMIPWwWHO | 


DIOP WWHOS 


DOIN WWH OS OWVIOURWWHO 


CS OO =2 C2 OTH CO CO ES 


WHO 


DIR OAC 


24029 
24134 
24238 
24342 
24446 
24549 
24651 
24754 
24855 
24957 
25058 
25159 
25259 
25359 
25458 
20557 
25656 
25755 
25853 
25950 
26048 
26145 
26241 
26337 
26433 
26529 
26624 
26719 
26813 
26908 


27001 
27095 
27188 
27281 
27373 
27466 
27557 


27649 
27740 
27831 
27922 
28012 
28102 
28192 
28281 
28370 
28459 
28547 
28635 
28723 
98811 
28898 
QB9R5 
29072 
29158 
29244 
29330 
29416 
29501 
29586 


24363 


24466 


24569 
24672 
2477. 

24876 
24977 
25078 
25179 
25279 
25379 
25478 
25577 
25676 


Sei 
257°C¢ 


25872 
25970 


26067 
26164 
26260 
26357 
26452 
26548 
26643 
26738 
26882 


26926 


27020 
27114 
27207 
27299 
27392 
27484 
27576 
27667 
27758 
27849 
27940 
28030 
28120 
28209 
28299 
28388 
28476 
28565 
98653 
28741 
28828 
28915 
29002 
29089 
29175 
29261 
29347 
29483 
29518 


29603 


24071 
24176 
24280 
24384 
24487 
24590 
24692 
24794 
24896 
24997 


25098 
25199 
25299 
25399 
25498 
25597 
25696 
25794 
25892 
25989 


26086 
26183 
26280 
26376 
26472 
26567 
26662 
26757 
26851 
26945 


27039 
27132 
27225 
27318 
27410 
27502 
27594 
27685 
orrent 


wlidd 
27867 
27958 
28048 
28138 
28227 
28317 
28405 
28494 
28582 
28670 
28758 
98846 
28933 
99020 
29106 
99192 
29278 
29364 
29450 
29535 


£9620 


24092 
24197 
24301 
24404 
24508 
24610 
247138 
24815 
24916 
25018 


25118 
25219 
25319 
25419 
25518 
25617 
25715 
25813 
25911 
26009 


26106 
26203 
26299 
26395 
26491 
26586 
26681 
26776 
25870 
26964 


27058 
27151 
27244 
27386 
27429 
27521 
27612 
27704 
27795 
27885 
27976 
28066 
28156 
28245 
283384 
28423 
28512 
28600 
28688 
28776 
28863 
28950 
29037 
29124 
29210 
29296 
29381 
29467 
99552 


29637 


24113 
RAR? 
24321 
24425 
24528 
24631 
24733 
24835 
24937 
25038 


25138 
25239 
25339 


25931 


26028 


26125 
26222 
26318 
26414 
26510 
26605 
26700 
26795 
26889 
£6983 

TO76 
27169 
27262 
27355 
27447 
27589 
27631 
alae 
27813 
27904 
27994 
28084. 
28174 
28263 
28352 
28441 
28529 
28618 
28706 
28793 


28881 
88968 
29054 
29141 
29227 
29313 
29398 
29484 
99569 


29654 


308 


) 


WOWOWOHOHNHDH SDOOSOSOOOSCSS COSCOHH HEHEHE 


G0 00 G0 G0 G0 G6 GD 0D DD OM MMO MDMDMDwO 


| aS Beek peek pr eek fed frat freak fk peek ek bak fk rk fr fk fk eek fem foek fk fk peek fed fred fk Peek Pek ee ee ek DO VDMV AD WO AMMA WWAW WWM WW WWwiwe 
MINIT WII OW To ooo: oof ¢ ae 


TABLE XVI.—COEFFICIENT OF CORRECTION FOR TEMPERATURE. 


! | 
t+ 4 — 64° t-+¢t/ ~ 64°]! ttt — 64°]| l¢-+¢ — 64° | 
/ /) / — = / ae = 

Bed 900 at 900 bat 900 . Sa 900 

= 4. | 2 | 
20° 0489 65° | + .0011 110° | + .0511 || 155° 1011 

21 = .0478 || 66 0022 || 111 0522 ||. 156 .1022 : 

22 0467 7 0033 112 0533 157 .1033 | 

23 0456 68 0044 113 0544 || 158 1044 : 

24 . 0444 69 0056 114 0556 |) 159 . 1056 | 

25 0433 "0 0067 115 .0567 | 160- 1067 | 

26 . 0422 v4 0078 116 .0578 || 161 1078 | 

Q7 .0411 72 0089 117 .0589 |) 162 1089 

| 28 ~ .0400 ve 0100 118 0600 | 163 1100 | 
29 .0389 7. 0111 119 0611 164 1111 

30 0378 "5 .0122 120 + 0622 165 .1122 | | 

{ 31 — .0367 Vi + 0133 121 0633 || 166 + .1133 | 

uh 32 0356 U7 0144 122 0644 |) 167 1144 | 

in 33 0344 78 0156 123 .0656 |) 168 1156 | 
ih 34 .0333 ” 0167 124 .0667 || 169 1167 

35s .0322 || 80 0178 125 .0678 || 170 1178 : 

| 36 0311 81 .0189 126 .0689 || 171 1189 | 
Hail 37 .0300 82 .0200 127 0700 || 1% 1200 
Sa 38 0289 83 0211 128 0711 1% 1211 
ne | 39 0278 84 0222 129 .0722 74 . 1222 
hale 40 0267 85 0233 130 + .0733 || 1%5 .1233 
Ma 41 — .0256 86 4. (0244 131 0744 || 17 +. 1244 
A a 2 0244 87 0256 132 0756 || 177 .1256 
Hi a | 43 0233 88 .0267 133 0767 ted? 1267 
LT 44 0292.6 1 89 0278 134 0778 7 1278 
a 45 0211 90 0289 135 .0789 | 180 1289 

lH 46 -0200 91 -0300 || 136 0800 || 181 1300 | 

Ay .0189 92 0311 137 .0811 | 182 1311 
48 0178 93 . 0322 138 (822 |) 1838 .1322 
49 .0167 || 94 .0333 139 0833 |! 184 1333 
50 — .0156 95 0344 140 + .0844 || 185 1344 
51 0144 96 + .0856 141 0856 || 186 + .1356 
52 0133 97 0367 142 .0867 || 187 .1367 
53 .0122 98 037 143 .0878 |) 188 .1378 
54 0111 99 0889 144 .0889 || 189 .1389 
55 .0100 || 100 .0400 145 .0900 |; 190 1400 

56 0089 101 0411 146 0911 191 1411 | 

57 0078 102 0422 147 0922 |; 192 1422 | 

58 0067 103 0433 148 0933 || 193 1433 | 
59 0056 104 0444 149 .0944 || 194 1444 

60 0044 105 0456 150 + .0956 195 1456 | 

61 — .0033 106 + .0467 151 .0967 || 196 + .1467 | 

62 .0022 107 0478 152 .0978 || 197 1478 | 
63 .0011 || 108 0489 153 0989 198 1489 

64 .0000 — || 109 .0500 154 .1000 |} 199 1500 | 


Hii TABLE XVII.—CORRECTION FOR EARTH’S CURVATURE AND 
REFRACTION. §119. 


. L° | H°|| L° | He |] Lo | wo || Le | B° || Lo | we ||Miles| He 
i| 800 | .002 |; 1300} .035 || 2300] .108 || 3300| .223 || 4300! .379 || 1 ye 
400 | .003 |, 1400] :040 || 2400) '118 || 3400] 1237 || 4400| {397 |} 2 | 2/285 | 
tj 500 | .005 |) 1500; .046 '| 2500] 128 || 8500) .251 |/ 4500] .415 || 3 | 5.142 | 
| 600 | .007 |} 1600) .052 ;| 2600] .139 || 3600| |266 || 4600] 1434 || 4 | 9.141 
i] 700 | .010 || 1700; .059 }} 2700] .149 || 3700) .281 || 4700! .453 5 14.282 
800 ] .013 || 1800} .066 | 2800} .161 || 3800) .296 || 4800| 472 6 | 20.567 
900 | .017 || 1900] .074 || 2900) :172 || 3900) [312 || 4900| 1492 || 7 | 27.994 
Wii 1000 020 |} 2000} 082 || 8000) .184 || 4000} .328 15000] .512 | 8 36.563 
Wit 1100 | .025 | 2100} .090 |} 3100} .197 || 4100] .345 ||5100} .533 9 46.275 
| 1200 | .030 || 2200] .099 |].3200} |210 || 4200] 362 |} 5200] 554 || 10. | 57.130 | 


TABLE XVIIl.—COEF FICIENT FOR REDUCING INCLINED STADIA 
MENTS TO THE HORIZONTAL. § 224. 


° 


10 


MEASURE 


0’ 


1.000000 
. 999696 
. 998782 
997261 


995134 


992404 


989074 
-985148 
. 980631 
975528 
. 969846 
-963591 
. 956772 
.949396 


941473 - 


.933011 
924022 
.914517 


.904507 - 
.894003. - 
883020 - ; 


.871569 ~ 


.859667 
847326 


.834561 °° 


821390 
807826 
. 793888 
T9591 
764954 
- 749994 
TBA729 
. 719179 
. 703361 
687296 


.671002 ° 


.654500 
.637810 
. 620952 
. 603946 
586814 


.569576 
992253 
.534867 
.517438 


.499988 © 


10’ 


. 999992 
999586 
998571 
. 996949 
. 994721 
.991891 
. 988461 
. 984436 
.979821 
.974621 
. 968843 
. 962494 
. 955581 
.948113 
. 940100 
.931550 
922474 
. 912883 
. 902790 
.892206 
.881143 
.869617 
.857640 
.845227 
. 832394 
.819156 
.805529 
. 79152 

helio 
. 762483 
T4 7471 
(82157 
.716561 
. 700700 
. 684595 
. 668266 
.651731 
.635011 
.618127 
.601099 
.583948 


566694 
.549359 
.531964 
.514530 
.497079 


20" 


999967 
. 999459 
. 998343 
. 996619 
994291 
.991360 
987831 
. 983708 
. 978995 
.973698 
967824 
. 961380 
954375 
946815 
938711 
. 9380073 
920911 
911236 
.901060 
.890395 
879254 
867652 
855601 
843117 
.830215 
.816911 
803221 
789161 
TATAD 
. 760002 
144939 
72957 
. 718935 
.698033 
.681889 
665524 
. 648957 
. 632208 
.615299 
598248 
.581079 


.5638810 
546464 
529061 
.511622 


-494170 


30’ 40’ 
. 999924 .999865 
9993815 . 999154 
. 998098 997886 
99627¢ .995910 
993844 993381 
. 990814 . 990250 
987185 . 986522 
982963 982202 
978152 977294 
972759 . 971804 
. 966790 . 965739 
. 960252 .959107 
953153 .951916 
945502 944174 
.937309 .935891 
928582 927077 
919334 917742 
90957. . 907899 
899316 897558 
.888571 886733 
877352 8754387 
865674 . 863684 
853550 851487 
.840996 838862 
828025 825825 
814656 812890 
800903 (98575 
786783 784396 
772814 . 769870 
157518 755015 
142399 . 789850 
. 726989 . 724893 
. 7113802 . 708662 
695858 692677 
679176 676457 
662776 . 660023 
646177 . 643893 
629401 626588 
-612466 - . 609630 
595395 592537 
978207 575332 
.560924 558036 
943567 . 540668 
-526156 528251 
508714 505805 
.491261 488353 


50’ 


999789 
998977 
‘997557 
. 995531 
992901 
989670 
985843 
981424 
976419 
970833 
964673 
957948 
950664 
942831 
934459 
9255577 
916137 
906209 
895787 
884883 
873510 


861681 
849412 
836718 
823613 
.810113 
196236 
781998 
67416 
152509 
T37294 


721790 


706015 
689990 
.673733 
657264 
640604 
623772 
.606790 
589677 
572455 


555145 
537768 
520345 
502897 
485445 


TABLE XIX.—LOGARITHM OF COEFFICIENT FOR REDUCING _IN- 
CLINED STADIA MEASUREMENTS TO THE HORIZONTAL. § 224. 


a 0/ 10/ 20/ 30/ 47 | 50 
0° | 0.000000 | 9.999996 | 9.999985 | 9.999967 | 9.999941 | 9.999908 
1 9.999868 | .999820 | .999765 | .999702 | 999633 | .999555 
2 999471 | .999379 | .999280 | .999173 | .999059 | 998938 
3 .998809 | .998673 | .998529 | .998879 | 998220 | ‘998055 
4 997882 | .997701 | .997514 | .997318 | .997116 | 996906 
5 .996689 | .996464 | .996232 | .995992 | (995745 | 995491 
6° | 9.995229 | 9.994959 | 9.994683 | 9.994899 | 9.994107 | 9.993808 
@ .993501 | .993187 | .992866 | .9925387 | .992201 | 1991857 
8 .991506 | .991147 | .990780 | .990406 | .990025 | .989636 
9 .989240 | .988836 | .988424 | 988005 | 987579 | ‘987144 
10 -986703 | 986253 | 985797 | .985332 | 984860 | 1984380 
11°. | 9.983893 | 9.983398 | 9.982895 | 9.982885. | 9.981867 -| 9.981342 
12 .980808 | .980268 | .979719 | .979163 | 978599 | ‘978027 
13 977447 | 976860 | .976265 | .975663 | 975052 | ‘974434 
14 .973808 | .973174 | .972532 | .971888 | 971225 -| _970560 
15 .969887 | .969206 | .968517 | .967820 | (967116 | (966403 
AS 16° | 9.965683 | 9.964954 | 9.964218 | 9.963473 |. 9-962721 | 9.961960 
ig 17 961192 | .960415 | 959631 .958838 | 958037 | 957229 
Hi 18 .956412 | 955587 | .954753- | .9538912 | 953063 | |952205 
Lacie 19 .951339 | 950465 | .949583 | 948692 | ‘947793 | |946886 
i 20 945970 | .945047 | .944114 | 1943174 | [942025 | “941968 
ta 21° | 9.940802 | 9.939828 | 9.938345 | 9.987354 | 9.936355 | 9.935347 
Ne 22 .934330 | .9883805 | .932271 .931229 | 930178 | 929119 
HI 23 928050 | .926974 | .925888 | .924794 | [993691 | ‘go257 
Hil 24 .921458 | .920329 | .919191 .918044 | 916888 | 915723 
a 25 914549 | .913366 | .912175 | 910974 909764 | 908546 
Hd 26° | 9.907318 | 9.906081 | 9.904835 | 9.903580 | 9.902316 | 9.901042 
HT 27 .899759 | .898467 | .897166 | 895855 | 894535 | 893206 
| 28 .891867 | .890519 | .889161 | .ss7794 | ‘secd17 | ‘885031 
Hl 29 .883635 | .882230 | .880815 |. .879390 | 877956 | (876512 
pil 30 .875058 | .873594 | 87212 .870637 | .869144 | 867641 
i 31° | 9.866127 | 9.864604 | 9.868071 | 9.861528 | 9.959974 | 9.858411 
i 32 .856837 | .855253 | .853659 | .852054 | .850439 | _84ssi4 
can 33 847178 | .845532 | 1843876 | .842209 | 840531 | _838843 
ch 34 .887144 | 835434 | .833714 | .831982 | 830040 | “g2s498 
35 .826724 | .824949 | .823163 | .821367 | :819559 | _817740 
36° | 9.815910 | 9.814068 | 9.812216 | 9.810352 | 9.808476 | 9.806589 
37 .804691 | .802781 | .800860 | .798927 | 796982 | .795026 
38 .793058 | .791078 | .789086 | .787082 | (735066 | .78380388 
39 (80998 | .778946 | .776882 | .774805 | (772716 | ..770614 
40 768500 | .766374 | .764235 | 762083 | .759919 | *. 757742 
ANE 41° | 9.755552 | 9.753349 | 9.751133 | 9.748904 | 9.746662 | 9.744407 
att 42 742138 | .739857 | . 737561 (35253 | .732931 730595 
va 43 728246 | 2725883 | 1723506 | 1721115 | .718710 | 7162901 
at 44 713858 | 711411 .708950 | .706474 | .708988 | .701479 
Hi 45 9.698959 | 9.696425 | 9.693876 | 9.691313 | 9.688734 | 9.686140 


TABLE XX.—LENGTHS OF CIRCULAR ARCS: RADIUS = 1. 


Length, | 


0000048 
0000097 
0000145 
0000194 
0000242 
0000291 
0000339 
0000388 
0000436 
0000485 


.0000533 
.0000582 
.0000630 
.0000679 
.0000727 
.0000776 
. 0000824 
.0000873 
.0000921 
.0000970 


.0001018 
.0001067 
.0001115 
.0001164 
0001212 
0001261 
.0001309 
-0001357 
.0001406 
.0001454 


.0001503 
.0001551 
. 0001600 
.0001648 
.0001697 
.0001745 
.0001794 
.0001842 
.0001891 
.0001939 


.0001988 
. 0002036 
.0002085 
.0002133 
0002182 
.0002230 
0002279 
.0002327 
.0002376 
.0002424 


.00024'7% 

.0002521 
.0002570 
.0002618 
.0002666 
0002715 
0002763 
0002812 


. 0002860 
. 0002909 


92 


SO OFS OC 


Length. || Deg 
. 0002909 il 
.0005818 2 
.0008727 3 
.0011636 4 
.0014544 5 
0017453 6 
.0020862 ff 
.0023271 8 
. 0026180 9 
.0029089 10 
.0081998 11 
.0034907 ile 
.00387815 13 
.0040724 14 
.0043633 15 
.0046542 16 
.0049451 % 
.0052360 18 
. 0055269 19 
.0058178 20 
.0061087 21 
. 0063995 22 
. 0066904 23 
.0069813 24. 
0072722 25 
.0075631 26 
.0078540 Wi 
.0081449 28 
.0084358 29 
. 0087266 30 
.0090175 al 
.0098084 || 32 
.0095993 || 83 
.0098902 || 34 
.0101811 || 35 
.0104720 36 
.0107629 || 37 
0110588 |} 3 

.0113446 || 39 
.0116855 40 
.0119264 41 
OL2e173 «rede 
.0125082 43 
.0127991 44 
.0130900 45 
.0138809 46 
.0186717 {i 
.0139626 48 
.0142535 49 
.0145444 50 
.0148353 51 
.0151262 52 
0154171 53 
.0157080 54 
.0159989 55 
.0162897 56 
0165806 57 
.0168715 58 
.0171624 ieeog 
ies | 60 


pie pits 


Length. | Deg Length. 
0174533 61 | 1.0646508 
0349066 62 1.0821041 
0523599 63 | 1.0995574 
0698132 64 1.1170107 
0872665 |} 65 1.1844640 
1047198 66 1.1519173 
1221730 37 1.1693706 
1396263 68 1.1868239 
1570796 69 12042772 
1745329 " 1. 2217305 
1919862 "1 1. 2391838 
2094395 12 1. 2566371 
. 2268928 "3 1.2740904 
2443461 "4 12915436 
2617994 15 1.3089969 
2792527 "6 1.8264502 
2967060 rit 13489035 
| .8141593 78 1.8613568 
| 8316126 vi 1.8788101 
| .8490659 80 1.8962634 
| .8665191 81 1.4137167 
| .8889724 || 82 1.4311700 
.4014257 83° | 1.4486233 
4188790 84 1.4660766 
4362323 85 14835299 
| .4537856 86 1.5009832 
| 4712389 87 1.5184364 
| 4886922 88 | 1.5358897 
5061455 || 89 1.5533480 
5285988 || 90 1.5707963 ° 
| 5410521 91 1.5882496 
BB85054 92 1.6057029 
| .57%59587 93 1.6231562 
| 5934119 94 1.64060°5 
6108652 95 16580628 
6283185 96 1.6755161 
6457718 97 1.6929694 
6632251 98 1.7104227 
6806784 99 1.7278760 
6981317 100 1.7453293 
7155850 || 101 1. 7627825 
7880883 || 102 | 41.7809358 
7504916 103 1.7976891 
7679449 || 104 1.8151424 
TR5 5B9R2 | 105 1.8825957 
8028515 106 1.85 500490 
8202047 107 | 1.8675028 
8377580 || 108 | 41.8849556 
.8552113 || 109 | 4.9024089 
8126646 || 110 | 1.919862 
8901179 111 | 1,9373155 
B0T5112 || 112 1.9547688 
J posbods | 113 1 9722224 
9424778 {| 114 1.9896753 
9599311 | 115 20071286 
97738844 || 116 | 9 9945819 
9948377 | 117 9 04203852 
1.0122910 | 118 | 2.0594885 
| 1.0297443 || 119 | 2.0769418 
| 120 | 2.0943951 


TABLE XXI.--MINUTES IN DECIMALS OF A DEGREE. 


, 0” 10” 15" 20" 30" 40" | 45" Bo" | 
0 | .oo000 | 00278 | .00417 | .00556 |) .00833 | .01111 | .01250 | .01389 | 0 
1 | 01667 | .01944 | .02083.} .02222 || .02500 | 02778 | .02917 | .03055 | 1 
9 | |03333 | .03611 | .03750 | .03889 || .04167 | .04444 | .04583 | .04722 | 2 
3 | 05000 | .05278 | .05417 | .05556 || .05833 | .06111 | .06250 | 06389 | 3 
4 | (06367 | .06944 | .07083 | 07222 || .07500 | .07778 | .07917 | .08056 | 4 
5 | (08333 | 08611 | .08750 | .08889 || .09167 | 09444 | .09583°| .09722 | 5 
s | 10000 | .10278 | .10417 | .10556 || .10833 | .11111 | .11250! 111389 | 6 
~ | ‘41667 | .11944 | .12083 | .12222 || .12500 | .12778 | .12917 | .13056 | 7 
g | (13333 | 18611 | .13750 | .18889 || 14167 | .14444 | 114588.) 14722 | 8 
9 | 15000 | 15278 | .15417 | .15556 |} .15838 | .16111 | .16250 | .16389 | 9 
10 | 16667 | .16944 | .17083 | .17222 |) .17500 | .17778 | .17917 | .18056 | 10 
i 11 | 18383 | .18611 | .18750 | .18889 |) .19167 | 19444 | .19583 | .19722 | 11 
12 | .20000 | |20278 | .20417 | 20556 || .20833 | .21111 | .21250 | .21389 | 12 
13 | (21667 | .21944 | .22083 | .22222 || 22500 | 22778 | .22917 | 123056 | 13 


14 23333 | .23611 | .28750 | .23889 || .24167 | .24444 | .24583 | 124722 | 14 
15 25000 | .25278 | .25417 | .25556 || .25833 | .26111 | .26250 | .26889 | 15 
| 16 26667 | .26944 | .27083 | .27222 || .27500 | .27778 | .27917 | .28056 | 16 
A 17 28333 | .28611 | .28750 | .28889 || .29167 | .29444 |-.29583 | .29722 | 17 

Hh 18 -30000 | .80278.| .380417 | .380556 || .80833 | .381111 | .381250 | .31389 | 18 


‘tid 19 | .31667 | .381944 | .32083 | .32222 || .82500 | .82778 | .82917 | .83056 | 19 
| it 90 | .33333 | .33611 | .383750 | .383889 || .84167 | .34444 | .34583 | .34722 | 20 
Hae | 21 | .35000 | .35278 | .85417 | .85556 || .35833 | .36111 | .36250 | .36389 | 21 


HI 99 | (36667 | .36944,| .37083 | .37222 || .37500 | .87778 | .387917 | 138056 | 22 
i 93 | °39333 | 138611 | .38750 | .38889 || .39167 | .39444 | .39583 | .39722 | 23 
| 24 | |40000-| .40278 | .40417 | .40556 |) .40833 | .41111 | .41250 | 141389 | 24 


li a5 | 41667 | .41944 | .42083 | .42222 || .42500 | .42778 | .42917 | 43056 | 25 
iis | 96 | 43388 | |43611 | .43750 | .43889 || 44167 | .44444 | 144583 | 144722 | 26 
Hy 97 | '45000 | (45278 | .45417 | .45556 || .45838 | .46111 | .46250 | .46389 | 27 
i 98 | |46667 | .46944 | .47083 | 47222 || .47500 | .47778 | .47917 | .48056 | 28 


29 48333 | .48611 | .48750 | .48889 || .49167 | .49444 | .49583 | .49722 | 29 
30 .50000 | .50278 | .50417 | .54556 || .50833 | .51111 | .51250 | .51389 | 30 


<8 .51667 | .51944°} .52083 | .52222 || .52500 | .5277 .£2917 | .538056 | 31 
32 53333 | .53611 | .538750 | .53889 || .54167 | .54444 | .54583 | .54722 | 382 
33 .55000 | .55278 | .55417 | .55556 || .55833 | .56111 | .56250 | .56389 | 33 
34 56667 | .56944 | .57083 | .57222 || .57500 | .57778 | .57917 | .58056 | 34 
35 58333 | .58611 | .58750 | .58889 || .59167 | .59444 | .59583 | .50722 | 35 
36 .60000 | .6027 .60417 | .60556 || .60883 | .61111 | .61250 | .61389 | 36 

37 | .61667 | .61944 | .62083 | .62222 || .62500 | .62778 | .62917 | .68056 | 37 

38 63333 | .63611 | .63750 | .63889 || .64167 | .64444 | .64583 | .64722 | 38 

39 .65000 | .65278 | .65417 | .65556 || .65833 | .66111 | .66250 | .66389 | 39 

40 | .66667 | .66944 | .67083 | .67222 || .67500 | .67778 | .67917 | .68056 | 40 

41 68333 | .68611 | .68750 | .68889 || .69167 | .69444 | .69583 | .69722 | 41 

2 .70000 |. .70278, | .70417 | .70556 || .70833 | .71111 | .71250 | .713889 | 42 


Tne 43 71667 | .71944 | .72083 | .72222 || .72500 | .72778 | . 72917 | .73056 | 43 
| } 44 73333 | .73611 | .73750 | .73889 || .74167 | .'74444 | .74588 | .'74722 | 44 
Palit 45 75000 |. 7527 5417 | .75556 || .75888 | .76111 | .76250 | . 76389 | 45 
Aa 46 .76667 | .76944.] 77083 | .77222 |) .77500 | 27777 JTI9TT | £78056 | 46 

i] 7 78333 | .78611 ; .78750 | .78889 || .79167 | .79444 | .79588 | .79722 | 47 


it 48 .80000 | .80278 | .80417 | .80556 || .80833 | .81111 | .81250 | .81389 | 43 
Mi 49 81667 | .81944 | .82083 | .82222 || .82500 | .8277 .82917 | .83056 | 49 

50 .83333 | .83611 | .83750 | .83889 || .84167 | .84444 | .84583 | .84722 | 50 | 
51 .85000 | .8527 .85417 | .85556 || .858383 | 86111 | .8625 .86389 | 51 
52 86667 | .86944 | .87083 | .87222 || .87500 | .87778 | .87917 | .88056 | 52 
53 88333 | .88611 | .88750 | .88889 || .89167 | .80444 | .89583 | .89722 | 53 
i 54 .90000 |. .90278 | .90417 | .90556 || .908383 | .91111 | .91250 | .91389 | 54 
Hi 55 .91667 | .91944 | .92083 | .92222 |) .92500 | .92778 | .92917 | .93056 | 55 


Hy 56 .93333 | .93611 | .93750 | .93889 || .94167 | .94444 | .94583 | .94722 | 56 
HH | 57 .95000 | .95278 | .95417 | .95556 || .95833 | .96111 | .96250 | .96389 | 57 
| 58 .96667 | .96944 | .97085 | .97222 || .97500 | .97778 | .97917 | .98056 | 58 

59 .98333 | .93611 | .98750 | .98889 || .99167 | .99444 | .99583 | .99722 | 59 


AR OF) |EL08 15°-¢|, 20° || GesOTN - dome] ) 50a OR pos tare 
31d 


| | 
In. hag | oP tase 6 TSF O-eledOy yh 11 

\ | | | 

| eons BBS baie 

i Agneta Saas eee aoe a | : aig | ma! | | 
| | 0 _ |Foot! .0833) 1667} .2500) 3333 ..4167| 5000) .5833! 6667} .7500 8333] 9167 
1-32 | .0026) .0859! . 1693] . 2526) .38859) .4193) .5026) .5859! . 6693} .'7526| 8859) .9193} 
| 4-16 |.0052! .0885).1719| 2552) .3385) 4219] 5052) .5885/ 6719] 7552) .8385/ 9219) 
| 3-32 |.0078 .0911| 1745) 2578) .3411) 4245) .5078) 5911 .6745| .7578) .8411) 9245 
1-8 |.0104) 0938) 1771] .2604) 3438) 4271) .5104| .5938) .6771| .7604| 8438) 9271 
| 5-32 |.0130 .0964) 1797) .2630) 8464) 4297) 5130 5964! .6797| 7630, 8464) .9297 
3-16 |.0156! .0990) 1823] .2656) .3490! .4323| .5156| 5990) .6823) .7656) .8490) 9323 
7-32 | 0182). 1016) 1849] .2682) .3516) .4849] 5182) 6016) .6849] .7682| .8516) .9349 
Tt .0208| .1042) .1875| .2708) .3542! .4375| 5208) .6042| .6875! 7708] 8542! 9875 
9-32 | 0234 1068] 1901] .2734| .3568) .4401|.5234) .6068] .6001| .7’734/ .8568) .9401 
5-16 | .0260) 1094 1927] 2760) 8594) .4427/ 5260) . 6094) .6927) 7760) .8594 9427 
11-32 | 0286) .1120) 1953] .2786) 8620] .4453) .5286) .6120) .6953) . 7786 .8620) .9453 
8-8 |.0313) .1146) 1979] 2813] .3646) 4479! 5813) .6146] 6979] .7813| .8646) 9479 
13-32 | .0339| 1172) .2005].2839) .3672! 4505] .5339/ 6172) .7005| .7839) .8672! 8505 
7-16 | .0365} 1198) 2031) .2865] 8698) .4531] 5365) .6198) .7081| .7865) .8698) 9531 
15-82 |.0391| 1224) 2057) .2891| 3724) .4557|. 5391] 6224) .7057| .7891| .8724| .9557 
1-2 |.0417|.1250! 2083] .2917| 3750] .4583] .541’7| . 6250) .7083] .791'7| .8750| .9583 
17-82 | 0443) .1276] 2109] 2943) .3776) .4609] .5443) .6276] .7109] .7943) .8776| .9609 
9.16 |.0469) 1302) 2135] 2969) .3802) .4635} .5469) . 6302] .'7135) .'7969| .8802/ .9635 
19-32 | 0495) .1328) .2161) 2995) 8828) .4661] .5495) 6828) .7161] 7995) 8828) 9661 
5-8 | .0521) .1354| .2188] 8021) .8854) .4688} .5521| .6354/ .7188) .8021! .8854) .968= 
21-32 | 0547) .1380) .2214] .3047| 8880) .4714| .5547| .6380} .7214| .8047| .8880] .9714 
11-16 | 0573) .1406) 2240) 3073} .8906) .4740} .5573) .6406] .7240) .8073 .8906] .9740 
23-32 | 0599) 1432 .2266] 3099) 3932 .4766) 5599 6432) .7266) 8099 8932. 9766 
0625} .1458} . 2292} 3125) .3958) .4792) .5625) 6458) . 7292! .8125) 8958! .9792 


0807 


.0651) .1484) .2318) 
-0677'| .1510} .2344 
0703} . 1536} .2370) 
0729) .1563) .2396 
0755) . 1589) . 2422) 
0781) 1615) 2448) 


1641} 2474) 


TABLE XXII.--INCHES IN DECIMALS OF A FOOT. 


<olol} 
OL77 
.8203} . 
.8229| 
. 8255 
.O201). 
3807) 


3984 
.4010). 


4036! 
4063! 


4089} 


4115) 


4141} 


.4818 | 
4844) 
4870} .f 
.4896) 
.4922 
.4948 
4974 


5651 | 
5677 


703 


5129 


5055 


5781 
5807 


.6484 
.6510 
6536 
. 6563 
6589 
.6615 
. 6641 


7318 | 
7344 
7370 


7396) . 


- (422 
(448 | 
TATE 


8151 
8177 
8208 
8229) 
8255 
8281} .9115) 
8307 


8984 | 
“9010 
"9036 
9063) 
9089 


.9818 


.9896 
. 9922} 
| 9948) 
9141) .9974 


. 9844) 1: 
9870) 27 


15-8; 


1-2 
17-82 

9-16 
19-82 

5-8 
21-32 
11-16 
23-32 


3-4 


25-32 


29-32 
15-16 
31-32 


8 


9 | 10 | 11 


TAPLE XXIII.—SQUARES, CUBES, SQUARE ROOTS. 


| [ | 
| i 
No. |Squares.| Cubes. BS eae y | Cube Roots. 
1 1 1 1.0000000 1.0000000 
2 4 8 1.4142136 1.2599210 
3 9 Q7 1.'7320508 1.4422496 
4 16 64 20000000 1.5874011 
5 25 125 2. 2360680 1.7099759 
6 36 216 2 4494897 1.8171206 
v 49 | 343 °2. 6457513 1.9129312 
8 64.1 512 2. 8284271 20000000 
9 81 729 3.0000000 2.0800837 
10 - 100 1000 3.1622777 2.1544347 
11 121 1331 3.3166248 2. 2239801 
12 144 1728 3.4641016 2 2894286 
13 169 2197 8.6055513 2.3513347 
14 196 2744 8.741657 2.4101422 
15 225 3375 3.87:29833 24662121 
16 256 4096 4 0000000 2.5198421 
17 289 4913 4. 1231056 2.5712816 
18 324 5832 4 2426407 2 6207414 
19 361 6859 4. 3588989 2. 6684016 
20 400 8000 4. 4721360 2.'7144177 
g 44} 9261 4 5825757 2.'7589243 
99 484 10648 4.6904158 2. 8020393 
23 529 12167 4.958315 2. 8438670 
24 576 1382 4. 8989795 2.8844991 
25 625 15625 5 0000000 2. 9240177 
26 376 17576 5 0990195 2. $624960 
27 729 19683 5. 1961524 3.0000000 
28 784 21952 5 2915026 3 .0365889 
29 841 24389 5.3851648 3.0723168 
30 900 27000 5. 4772256 3. 1072325 
31 961 29791 55677644 3. 1413806 
32 1024 32768 5. 6568542 3.1748021 
33 1089 35937 57445626 8.2075343 
34 1156 39304 5. 8309519 3. 2396118 
85 1225 42875 59160798 3.2710663 
at! 36 1296 46656 6.0000000 3.3019272 
ait | °* 37 1369 50653 6. 0827625 3. 3322218 
att | 38 1444 54872 6.1644140 3.8619754 
a 39 1521 59319 6.2449980 3.3912114 
a 40 1600 64000 6.3245553 3.4199519 
a 41 1681 68921 64031242 3.4482172 
ii} 42 1764 74088 6 .4807407 3.4760266 
al 43 1849 79507 6.5574385 3.5033981 
Ht 44 1936 85184 6. 6332496 . 8.5808483 
i 45 2025 91125 6..7082039 35568933 
il 46 2116 97336 6..7823300 3.5830479 
Wii " 2209 103823 68556546 3.6088261 
48 2304 110592 6. 9282022 3 6342411 
' 49 2401 117649 7 0000000 3.6593057 
| 50 , 2500 125000 70710678 8.6840314 
51 2601 132651 % 1414284 3.7084298 
i 52 2704 140608 72111026 3.7325111 
| 53 2809 148877 72801099 8.7562858 
| 54 2916 157464 % 3484692 3.7797631 
55 3025 166375 74161985 3. 8029525 
) 56 3136 175616 74833148 3. 8258624 
i BY 3249 185193 7 5498344 3.8485011 
i 58 3364 195112 "6157731 38708766 
iF 59 3481 205379 76811457 3. 8929965 
60 3600 216000 % V459667 8.9148676 
61 3721 226981 78102497 3.9364972 
62 3844 238328 78740079 3.9578915 
OP ew . ee ee ee sa a = 


1 


| 
Reciprocals. 


1.000000000 
- 500000000 
8333838333 
. 250000000 

- 200060000 
. 166666667 
- 142857148 
- 125000000 
Meh mabol 


. 100000000 
-090909091 
083383833 
076923077 
071428571 
. 066666667 
. 062500000 
. 058823529 
055555556 
052631579 


- 050000000 
047619048 
045454545 
043478261 
041666667 
.040000000 
038461538 
-037037037 
085714286 
034482759 
. 033333333 
082258065 
031250000 
.030303030 
029411765 
628571429 


OTT 77 


027027027 
.026315789 
025641026 
025000000 
024390244 
023809524 
023255814 
022727273 
022222229 
021739130 
021276600 
020833333 
020408163 
020000000 
019607843 
019230769 
018867 925 
018518519 
.018181818 
017857143 
017545860 
.017'241379 
.016949153 
016666667 
016303443 
016129032 


31d 


CUBE ROOTS, AND RECIPROCALS. 


bot 
oO 


He Co OO Ret 


mJ AF HJ AF IF FI 


OO O22 OF 


| Squares. 


3969 
4096 
4225 
4356 
4489 
4624 
4761 
4900 
5041 
5184 
5329 
5476 
5625 
5776 
5929 
6084 
6241 


10404 
10609 
10816 
11025 
11236 
11449 
11664 
11881 
12100 
12321 
12544 
12769 
12996 
13225 
13456 
13689 
13924 
14161 


14400 
14641 
14884 
15129 
15376 


Cubes. 


250047 
262144 
274625 
287496 
300763 
3144382 
328509 
343000 
857911 
373248 
3889017 
405224 
421875 
438976 
456533 
474552 
493039 
512000 
531441 
551368 
571787 
592704 
614125 
636056 
658503 
681472 
704969 
729000 
753571 
773688 
804357 
830584 
857375 
884736 
912673 
941192 
970299 


100000 

1030301 
1061208 
1092727 
1124864 
1157625 
1191016 


1225043" 


1259712 
1295029 
1331000 
1367631 
1404928 
1442897 
1481544 
1520875 
1560896 
1601613 
1643032 
1685159 

728000 
1771561 
1815848 
1860867 
1906624 


Square 
koots. 


9372539 
0000000 
0622577 
1240384 
1853528 
2462113 
.8066239 
3666003 
4261498 
4852814 
5440037 
6028253 
6602540 
T177979 
7749644 
8317609 
8881944 
9442719 
0000000 
0553851 
1104336 
1651514 
2195445 
2736185 
38273791 
.8808315 
.4339811 
.4868330 
5393920 
.5916630 
.6436508 
6953597 
7467943 
7979590 
8488578 
8994949 
9.9498744 


10.0000000 
10.0498756 
10.0995049 
10.1488916 
10.1980390 
10.2469508 
10.2956301 
10.3440804 
10.3923048 
10.4403065 


10. 4886385 
10.5356538 
10.5830052 
10.6801458 
10.¢770783 
10.7238053 
10.7'703296 
10.8166538 
10.8627805 
10.9087121 


10. 9544512 
11.00 10000 
11.0453610 
11.0905365 
11.1855287 


Mmmm anm- 


316 


Cube Roots. 


3.979057 
4.0000000 
4.0207256 
4.0412401 
4.0615480 
4,0816551 
4,1015661 
4.1212853 
4.1408178 
4.1601676 
4.1798390 
4.1988364 
4.2171633 
4 2358236 
4 2543210 
4 2726586 
4.2908404 
4 3088695 
43267487 
8444815 
3620707 
8795191 
3968296 
4140049 
4310476 
79602 
4647451 
4814047 
.4979414 
.5143574 
.5306549 
5468359 
. 5629026 
5788570 
.5947009 
.6104363 
.6260650 
.6415888 
6570095 
.6723287 
.6875482 
. 7026694 
. 7176940 
. 7326235 
~7474594 
. (622032 
. 7768562 
.'7914199 
.8058955 
.8202845 
.$345881 
.8488076 
.8629442 
.8769990 
.8909732 
9048681 
9186847 
9324242 
9460874 
9596757 
9731898 
9866310 


Reciprocals. 


015873016 
.015625000 
.015384615 
.015151515 
.014925373 
.014705882 
.01449275 

.014285714 
.014084507 
013888889 
.013698630 
.013518514 
013333333 
.013157895 
012987013 
012820513 
01265822: 


012500000 
012345679 
-012195122 
.612048193 
.011904762 
.011764706 
.011627907 
.011494253 
.011363636 
011235955 


011111111 
.010989011 
.010869565 
.010752688 
. 010638298 
.010526316 
.010416667 
.010309278 
.010204082 
.010101010 


.010000000 
009900990 
009803922 
009708738 
009615385 


.009523810 
.009433962 
.009345794 
005259259 
.0091743812 
.009090909 
_ .009009009 
.0089285; 1 
.008849558 
.008771930 
. 008695652 
.008620690 
.008547009 
.008474576 
.008403361 
008333385 
.008264463 
.008196721 
.008130081 
.008064516 


aa eee 


Squares, 


15625 
15876 
16129 
16384 
16641 
16900 
17161 
17424 
17689 

7956 
18225 
18496 
18769 
19044 
19321 


19600 
19881 
20164 
20449 
20736 
21025 
21316 
21609 
21904 
22201 


22500 
22801 
23104 
23409 
23716 
24025 
24336 
24649 
24964 
25281 
25600 
25921 
26244 
26569 
26896 
27225 
27556 
27889 
28224 
28561 
28900 
29241 
29584 
29929 
30276 
30625 
30976 
31329 
31684 
32041 

32400 
32761 

33124 
33489 


33856 
84225 
34596 


Cubes. 


1953125 
2000376 
2048383 
2097152 
2146689 
2197000 
2248091 
2299968 
2352637 
2406104 
2460875 
2515456 
2571353 
2628072 
2685619 
2744000 
2803221 
2863288 
2924207 
2985984 
8048625 
8112136 
3176523 
8241792 
8307949 
8375000 
3442951 
8511808 
3581577 
8652264 
8723875 
3796416 
38869893 
3944312 
4019679 


4096000 
4173281 
4251528 
4330747 
4410944 
4492125 
4574296 
4657463 
4741682 
4826809 
4913000 
5000211 
5088448 
S1LTTT17 
5268024 
5859375 
5451776 
5545233 
5639752 
5735339 
5832000 
5929741 
6028568 
6128487 
6229504 
6331625 
6434856" 


Square 
Roots. 


11.1803399 
112249722 
11.2694277 
11.8137085 
11.8578167 
11.4017543 
11. 4455231 
11.4891253 
115325626 
115758369 
11.6189500 
11.6619038 
11. 7046999 
11.7473401 
11.7898261 
11.8321596 
11.874342 

119163753 
11. 9582607 
12.0000000 
12.0415946 
12.0830460 
12.1248557 
121655251 
22065556 
2. 2474487 
12.2882057 
12 38288280 
12.3693169 
124096736 
12.4498996 
12.4899960 
12..5299641 
12.5698051 
12. 6095202 


2.6491106 
12.6885775 
12. 7279221 
12.7671453 
128062485 
128452326 

2.8840987 
12.9228480 

2.9614814 
13 0000000 


13..0334048 
13.0766968 
13.114877 

13. 1529464 
13. 1909060 
13. 2287566 
132664992 
13. 3041347 
13.3416641 
133790882 
13.4164079 
13. 4536240 
13.4907376 
13.5277493 
13.5646600 
136014705 
136381817 


TABLE XXIII—SQUARES, CUBES, SQUARE ROOTS. 


Cube Roots. 


5.0000000 
5.0132979 
5 .0265257 
5.0396842 
5.0527743 
5.0657970 
5.0787531 
5. 0916434 
1044687 
1172299 
1299278 
. 1425632 
1551867 
. 1676493 
1801015 


1924941 
2048279 
5. 2171034 
52293215 
5.2414828 
5.2585879 
5.2656374 
5. 2776821 
5. 2895725 
3014592 


8132928 
.8250740 
. 83868033 
. 38484812 
. 8601084 
53716854 
5, 8882126 
5 3946907 
54061202 
5.4175015 
5. 4288352 
5. 4401218 
5.4513618 
5 .4625556 
5.4737037 
5. 4848066 
5.4958647 
5 5068784 
55178484 
5 5287748 
5.5396583 
.5504991 
.5612978 
5720546 
5827702 
.59384447 
6040787 
5.6146724 
6 .6252263 
5. 6357408 
5 6462162 
5.656652 

5.6670511 
56774114 
5.6877340 
56980192 


Or 


Crorororoer 


OU OT Or Or OT OC 


10817 


5. 7082675 


Reciprocals. | 


.008000000 
-007936508 
007874016 
007812500 
; 7751938 
007692308 
7633588 
007575758 
-007518797 
007462687 
007407407 
-007352941 
-007299270 
007246377 
007194245 
007142857 
007092199 
007042254 
006993007 
006944444 
- 006896552 
006849315 
- 006802721 
006756757 
.006711409 
006666667 
-006622517 
006578947 
006535948 
- 006493506 
-006451613 
.006410256 
-006369427 
006329114 
006289308 


- 006250000 
-€06211180 
-006172840 
006134969 
-006097561 
- 006060606 
. 006024096 
- 005988024 
-005952381 
-005917160 


.005882353 
005847953 
005818953 
005780347 
005747126 
005714286 
.005681818 
-005649718 
005617978 
005586592 


005555556 
005524862 
005494505 
005464481 
005434783 
-005405405 
-005376344 


CUBE ROOTS, AND RECIPROCALS. 


| _ 
No. | Squares. Cubes. eee Cube Roots, | Reciprocals. 
—$——— |—_—__—_———_- -_—_ —— — —- 
187 34969 6539203 13.6747943 5.7184791 005847594 
188 35344 6644672 13.7113092 5.7286543 .005319149 
189 85721 3751269 13. 7477271 5.7387936 005291005 
190 36100 6859000 13.7840488 5.7488971 .005263158 
| 191 36481 6967871 13.8202750 5.7589652 005235602 i 
192 36864 (0778 13 .8564065 5. 7689982 005208333 
| 193 37249 7189057 -13.8924440 5.7788966 .005181347 | 
194 376386 7301384 13. 9283883 5.'7889604 .005154639 } 
195 | 388025 TA14875 13.9642400 5.'7988900 .005128205 i og 
| 196 38416 7529536 14.0000000 5. 8087857 005102041 Will 
197 38809 7645373 14.0356688 5.81 ee 005076142 i 
198 | 39204 7762392 14.0712473 5 828476 .005050505 
199 39601 7880599 14, 1067360 5.83827 25 . 005025126 
200 40000 8000000 14.1421356 58480355 005000000 
: 201 40401 8120601 14.1774469 5. 8577660 004975124 i 
202 40804 8242408 14.2126704 5.8674643 604950495 i 
203 41209 8365427 14. 2478068 5.8771307 004926108 | 
204 41616 8489664 14. 2828569 5. 8867653 004901961 Mi 
205 42025 8615125 14.3178211 5 8963685 004878049 i 
| 206 42436 741816 =| 14.3527001 5.9059406 004854369 I 
207 42849 8869743 14. 3874946 59154817 004830918 i 
208 43264 8998912 14. 4222051 5. 9249921 004807692 i} 
209 43681 9129829 14. 4568323 5. 9344721 004784689 \ 
210 | 44100 9261000 14.4913767 5 9439220 .004761905 
211 44521 9393931 14.5258390 5.95388418 .004739336 i 
212 44944 9528128 14.5602198 5. 9627320 004716981 i 
213 45369 9663597 14.5945195 | 5.9720926 .004694836 i 
214 45796 9800344 14.6287888 | 5.9814240 .004672897 i 
215 46225 9938875 14. 6628783 5.9907264 .004651163 i 
i 216 46656 10077696 14.69693885 6.0000000 . 004629630 ; 
217 47089 10218313 14.7309199 6.0092450 .004608295 ! 
218 | 7524 10360232 14.764828 6.0184617 .004587156 i 
219 47961 10503459 14.7986486 6 .0276502 004566210 ' 
22 48400 10648000 14.8823970 |  6.0868107 004545455 
eal 48841 10798861 14.8660687 |  6.0459435 004524887 
222 49284 10941048 14.8996644 | 6.0550489 .004504505 
e2% 49729 11089567 14.9831845 | 6.064120 004484305 
224 50176 11239424 14.9666295 | 6.0731779 -004464286 | 
225. | 50625 11390625 15. 0000000 6 .C8E2020 004444444 Wa 
226 51076 11548176 15 .0882964 6.0911994 -004424779 | 
227 51529 11697083 150665192 6.1001702 .004405286 
22 51984. 11852852 15 .0996689 6.1091147 .004825965 3 
229 52441 12008989 15.1827460 61160832 004866812 
| 
230 52900 12167000 15.1657509 6. 1269257 .004 34 7826 | 
231 53361 12326391 15.1986842 6.1857924 004229004 | 
232 53824 12487168 15. 2315462 6.1446837 "004310845 
233 54289 12649337 15 .2648375 6.184495 004291845 | 
234 54756 12812904 15.29705 85 6.1622401 .00427 8504 
2395 55225 12977875 15.8297097 6.1710058 : 04D5E 319 
236 55696 13144256 15.3622915 6.1797466 004237288 
237 56169 13312053 15 38948043 6.1&84628 .004219409 | 
238 56644 13481272 15 4272486 6.1971544 -004201681 
239 57121 18651919 15 .4596248 6.2058218 .C04184100 | 
: 240 57600 13824000 | 15.491933: 6.2144650 004166667 
2A1 58081 18997521 =| 15.5241747 6. 2280843 .C04149878 
242 58564 14172488 15 5563492 6.2316797 -004182281 
243 59049 14348907 15.5884573 6. 2402515 004115226 a 
244 59536 | 14526784 15 6204994 6 .2487998 . 004098361 
245 60025 14706125 15 .6524758 6 .2573248 .004081683 
246 60516 14886936 15.6843871 6. 2658266 .004065041 
247 61009 15069223 15.7162836 6.2743054 004048583 
| 248 61504 15252992 15.7480157 6.2827613 .004032258 


318 


No, Squares. Cubes. 
—— oe. hese 
249 62001 15438249 
250 62500 15625000 
251 63001 15813251 
252 63504 16003008 
253 64009 16194277 
254 64516 16387064 
255 65025 16581375 
255 65536 16777216 
257 66049 16974593 
258 66564 17173512 
259 67081 17373979 
260 67600 17576000 
261 68121 17779581 
262 68644 17984728 
263 69169 18191447 
264 69696 18399744 
265 70225 18609625 
266 10756 18821096 
| 267 71289 19034163 
268 71824 19248832 
At hg 269 72361 19465109 
iil 270 72900 19683000 
Wet 271 73441 19902511 
Ne 272 73984 20123648 
Hy 273 %4529 20346417 
Hi Q74 75076 20570824 
iii! Q75 75625 20796875 
276 "6176 21024576 
277 76729 21252933 
278 77284 21484952 
279 V7841 21717639 
280 78400 21952000 
281 78961 22188041 
282 9524 22425768 
283 80089 22665187 
284 80656 22906304 
285 81225 23149125 
286 81796 23393656 
287 82369 23639903 
288 82944 23887872 
289 83521 24137569 
290 84100 24389000 
291 84681 24642171 
292 85264 24897088 
293 85849 25153757 
294 86435 25412184 
295 87025 25672375 
296 87616 25934336 
297 | 88209 26198073 
298 88804 26463592 
299 89401 26730899 
300 90000 27000000 
301 90601 27270901 
302 91204 27543608 
303 91809 27818127 
304 92416 28094464 
805 93025 28372625 
306 93636 28652616 
307 94249 28934443 
308 94864 29218112 
309 95481 29503629 
310 96100 29791000 


7 
q 

‘17.0880075 
‘ 


Square 
Roots. 


15.779733 
15.8113883 
158429795 
158745079 
159059737 
15.9373775 
15.9687194 
160000000 
16.0312195 
16, 0623784 
160934769 


16. 1245155 
16. 1554944 
16. 1864141 
16..2172747 
16.2480768 
16.2788206 
16 .3095064 
163401346 
16.3707055 
16.4012195 
164316767 
16.4620776 
164924225 
16.5227116 
16. 5529454 
165831240 
166132477 
16. 6433170 
16..6733320 
16. 7032931 
167332005 
16.7630546 
16. 7928556 
168226038 
168522995 
168819430 
16.9115345 
16. 9410743 
169705627 
70000000 
0293864 
0587221 


1172428 
171464282 
7.1755640 
172046505 
7,2336879 
72626765 
@.2916165 
78205081 
7 .8493516 
7.3781472 
7 .4068952 
7 .4855958 
74642492 
74928557 
75214155 
75499288 
7.5788958 
7. 6068169 


319 


TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS, 
SS ee ae eee 


Cube Roots, 


62911946 


6.2996053 
6.3079935 
6.3163596 
6.38247035 
6. 3330256 
6.3413257 
6 .3496042 
68578611 
68660968 
6.3743111 
6.3825043 
6.3906765 
6.3988279 
6. 4069585 
6.4150687 
6.4231583 
6 .4312276 
6.4392767 
6.4473057 
64553148 


6.4633041 
6.4712736 
64792236 
6.4871541 
6.4950653 
5029572 
.5108300 
5186839 
5265189 
.9343351 
. 0421326 
9499116 
.55716722 
.9654144 
5731885 
. 5808443 
5885323 
5962023 
6038545 
.6114890 
.6191060 
. 6267054 
6342874 
6418522 
6493998 
6569302 
6644437 
6719403 
6794200 
. 6868831 
6943295 
7017593 
7091729 
7165700 
7239508 
7318155 
7386641 
7459967 
6 .7583134 
6.7606143 
6.7678995 


De D2 ID 


Reciprocals, 


.004016064 


004000000 
.003984064 
003968254 
.003952569 
.003937008 
003921569 
003906250 
.003891051 
003875969 
. 003861004 


003846154 
. 003831418 
003816794 
003802281 
.003787879 
.003773585 
.003759398 
003745818 
0037313843 
008717472 
003703704 
0038690037 
.003676471 
.003663004 
. 003649635 
. 003636364 
. 003623188 
. 003610108 
003597122 
- 003584229 


.003571429 
.003558719 
. 003546099 
. 003533569 
.008521127 
.0035087'7% 
.0034965038 
.0038484321 
.003472222 
- 003460208 
.003844827' 

. 003436426 
- 003424658 
.003412969 
.003401361 
.003389831 
.003378378 
. 003367003 
.003355705 
. 003344482 


.0033338333 
. 003322259 
.003311258 
. 0038800330 
. 008289474 
. 0038278689 
. 0038267974 
. 0038257329 
. 003246753 
.098286246 
. 003225806 


} 
CUBE ROOTS, AND RECIPROCALS, 
| - 
| | | | 
No. | Squares, Cubes. = aoa | Cube Roots. : Reciprocals, 
Atal | i Been, (ORE OE 
Goat: aes 96721 | 80080231 | 176351921 6.751690 Seed ios, 
312 97344 30371828 17..6635217 6. 7824229 . 003205128 
313 97969 30664297 176918060 6.7896613 . 003194888 
314 98596 30959144 177200451 6.7968 844 003184713 i 
815 99225 31255875 17 . 7482393 68040921 .003174603 | 
316 99856 31554496 17.7763888 6.811247 | .008164557 
317 100489 | 81855013 17 .8044938 6.8184620 . 008154574 } 
318 101124 82157432 =| 178325545 6.8256242 Oates 4. | 
319 101761 32461759 7.8605711 6.832714 .003134796 
f 
320 102400 32768000 178885438 6.8399037 Sotto Hit: 
321 103041 33076161 | 17.9164729 6.8470213 | 003115265 
322 103684 33386248 7.9443584 6.8541240 . 063105590 | 
323 104329 33698267 179722008 68612120 .002095975 
S24 104976 34012224 | 18.0000000 6. 8682855 003 B64: 20 ) 
825. | 105625 34328125 18.0277564 6.8753443 003071692 i 
826 | 106276 34645976 18 0554701 6. 8823888 003067485 
B27 | ~—:106929 34965783 18.0831413 |  6.8894188 .003058104. \ 
828 | 107584 85287552 18. 1107703 6.8964345 .003048780 | 
329 108241 35611289 18, 1383571 6. 9034359 “003039514 i 
330 108900 35937000 18. 1659021 6.910422 .003030303 | 
33 109561 36264691 181934054 6.9173964 .003021148 i 
332 | 11022 36594368 18 . 2208672 6.9243556 | 003012048 i 
333 | 110889 86526037 18. 2482876 6.9318008 002003003 i 
834 111556 37259704 18. 2756669 6 9382321 . 002994012 i 
330 112225 875953875 18.8030052 6.9451496 .C02985075 | 
| 336 112896 37933056 18 .3303028 6 .9520533 .002976190 
337 113569 88272753 18.3575598 6.9589484 | _002967359 i 
338 114244 38614472 18 .3847763 6.9658198 | .002958580 
83! 114921 38958219 18.4119526 6. 9726826 .002949853 | 
115600 39304000 18.4390889 6.9795321 . 002941176 
116281 | 39651821 18466185: 6. 9863681 002982551 
116964 | 40001688 18. 4932420 6. 9931906 .002923977 
117649 | 40353607 18.5202592 7 0000000 .0029154 452 
: 118336 | 40707584 185472870 7 0067962 . 00290697’ 
119025 | 41063625 18.5741756 7 0035791 "002898551 
119716 | 41421736 18. 6010752 70203490 .002890173 
120409 41781923 18. 6279360 #0271058 002881844 
121104 | 42144192 18.6547581 7.0338497 .002873563 
121801 |. 42508549 18.6815417 70405806 .002865830 
122500 42875000 18.7082869 7 .0472987 0088571438 i 
123201 43243551 18.7349940 7.054004] . 002849003 ( 
123904 | 43614208 18. 7616630 70606967 . 002840909 | 
124609 | 43986977 | 18.7882942 70673767 | 002832861 tH 
35 125316 44361864 18.8148877 7.0740440 | 002824859 | 
3855 126025 44738875 18.8414437 70806988 .002816901 i 
356 126736 45118016 18.8679623 70873411 . 002808989 
357 127449 45499293 18.8944436 7.0939709 . 002801120 | 
358 128164 45882712 18.9208879 71005885 .002793296 
359 128881 46268279 18.9472953 71071937 002785515 
360 129600 46656000 18. 9736660 71137866 0027777 78 
361 130321 47045881 19.0000000 V. 1203674 "00277008 
362 131044 47437928 19 .0262976 71269360 .002 + ODA3t 
| 363 131769 | 47882147 190525589 7. 1334925 002754821 
364 182496 | 48228544 19.0787840 7. 1400370 002747253 
| 865 133225 48627125 19.1049732 71465695 .002739726 
| 366 33956 49027806 19.1311265 7.1580901 "002732240 
367 | 134689 49430863 19.1572441 7’ .1595988 002724796 
868 | 135424 49836032 19. 1833261 71660957 002717391 
369 136161 50243409 192093727 7.1725809 .002710027 
370 136900 50653000 192353841 71790544 .002702703 
37 37641 51064811 19.2613603 71855162 .002695418 
372 138384 51478848 19.2873015 7.1919663 .002688172 


L bees Ae get et a 
520 


873 | 189129 
374 | 139876 


| 75 140625 

| 346 | 141876 

377 | 142129 

378 | 142884 

379 | 143641 

380 | 144400 

381 | 145161 

p> 882 | «145924 

; 883 | 146689 

| 884 | 147456 

| 885 148225 

| 386 148996 

| 387 | 149769 

888 | 150544 

889 | 151821 

390 | 152100 

vii 391 | . 152881 

| 892 .| 153664 

li | 398 | 154449 

lit | 394 155236 

Hn | 895 | 156025 

HH 396 | 156816 

Hi | 397 | 157609 

iy } 398 158404 

Na | 399 159201 
fen | 

| 400 | 160000 

401 160801 

402 161604 

403 162409 

| 404 163216 

405 | 164025 

| 406 164836 


407 165649 
408 166464 
409 167281 


| 410 | 168100 
| 411 | 168921 
WW 412 | 169744 
vit 413 | 170569 
i 414 | 171396 
Alt 415 | 172225 
A 416 | 173056 
a 417 | 173889 
ii 418 | 174724 


419 | 175561 
420 | 176400 
177241 
422 | 178084 
423 | 178929 
424 | 179776 
425 | 180625 


426 181476 
42% 182529 


428 183184 
429 184041 
430 184900 
185761 
432 186624 

| 187489 
434 | 188856 


| 


| Squares. | 


Cubes. 


51895117 
52313624 
52734375 
58157376 
53582633 
54010152 
54439939 
54872000 
55306341 
55742968 
56181887 
56623104 
57066625 
7512456 
7960603 
58411072 
58863869 
59319000 
59776471 
60236288 
60698457 
61162984 
61629875 
62099136 
32570773 
63044792 
63521199 
64000000 
64481201 
64964808 
65450827 
65939264 
66430125 
66923416 
67419143 
67917312 
68417929 
68921000 
69426531 
69934528 
70444997 
70957944 
1473875 
71991296 
2511713 
73034632 
#3560059 
4088000 
74618461 
75151448 
"5686967 
36225024 
"6765625 
77308776 
7854483 
78402752 
78953589 
79507000 
80062991 
80621568 
81182737 
81746504 


Square 
xn te | Cube Roots, 
19.3132079 |  %.1984050 
19.3390796 | 7.204882 
19.8649167 | 7.2112479 
19.3907194 |  7.2176522 
19.4164878 |  7.2240450 
194422221 |. 7.2304268 
19.4679223 | 72367972 
19.4935887 | '%.2431565 
19.5192218 | 7.2495045 
19.5448203 | 7.2558415 
195703858 72621675 
195959179 % 2684824 
19 .6214169 7 2747864 
19.6468827 7 2810794 
196723156 1 2873617 
19.6977156 72936330 
19. 7230829 |  7.2998986 
197484177 | 7.3061436 
19.7737199 | %.3123828 
19.7989899 | 7%.3186114 
19.8242276 |  %.3248295 
19. 8494332 7 .8310369 
19.8746069 % 3372339 
19.8997487 | 7,3484205 
19.9248588 | 17.3495966 
19.9499373 | %.8557624 
19.9749844 | %.8619178 
20.0000000 | %.8680630 
20.0249844 | 7. 8741976 
20.0499377 | 7.8808227 
20.0748599 % 2864373 
20.0997512 7.3925418 
20. 1246118 7 3986363 
20.1494417 | 7.4047206 
20.1742410 |  7.4107950 
20.1990099 |  7.4168595 
20. 2237484 7 4229142 
20. 2484567 7 4289589 
202731349 7 4349938 
20.2977831 74410189 
203224014 4470342 
20. 3469899 74580399 
203715488 74590359 
20.3960781 7 4650223 
20.4205779 7.4709991 
20.4450483 7 4769664 
20.4694895 74829242 
204939015 7. 4888724 
20 5182845 74948113 
205426386 75007406 
20.5669638 |  7.5066607 
20 5912603 75125715 
20.6155281 75184730 
20. 6397674 7 5243652 
20.6639783 75302482 
20. 6881609 7 5361221 
20.7123152 75419867 
20.7364414 7 5478423 
207605395 7 5536888 
20.7846097 75595263 
20. 8086520 75653548 
7.5711743 


20. 8326667 


321 


TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS, 


| 


Reciprocals. 


002680965 
002673797 
002666667 
002659574 
002652520 
002645503 
002638522 


.002681579 
. 002624672 
.002617801 
.002610966 
.002604167 
.002597403 
. 002590674 
002583979 
002577320 
002570894 
002564108 
.002557545 
.002551020 
002544529 
002588071 
. 002531646 
. 002525253 
.002518892 
002512563 
002506266 


002500000 
.002493766 
.002487562 
-002481390 
002475248 
002469136 
. 002463054 
002457002 
002450980 
.002444988 


00243902 
002433090 
002427184 
- 002421308 
002415459 
002409639 
002403846 
002398082 
002392344 
002386635 
002380952 
002375297 
002369668 
002364066 
002358491 
002352941 
002347418 
- 002341920 
002336449 
002331002 
002325581 
002320186 
002314815 
.0023809469 
002304147 


CUBE ROOTS, AND RECIPROCALS. 


Sta ae Werke “Be pee Ralaeen ae 
| | | ] 
| a 4 = t; Square > | 
: No. (Squares. | Cubes, | Sheil | Cube Roots. | Reciprocals. 
435 | 189225 82312875. |  20.8566536 7.576984: 20885) 
436 | 190096 | 82881856 | 20. 8806130 apes | Prgsee 
|. 437 | 790969 | 83453453 |  20.9045450 | 7.5885793 “002258330 
438 | 191844 84027672 | — 20.9284495 75043633 | 7002985105 
439 | 19z721 | 84604519 | 20.9523268 | 7.600185 | 02277904 | 
| 440 | 193600 | 85184000 | 20.9761770 | 7.6059049 |  .o02272727 | 
| 440 | 133600 | Beresisn | 2120000000 | y-6146626 | |. onaabzarG | 
: 442 | 195364 | 86350888 | 21.0237960 | 7.6174116 | “026243 | 
443 | 196249 | 86038307 | 21.0475652 | 716231519 | [002257336 | 
: 444 |< 197136 Wepsse, | 21.0718075 | 7.688887 |  “oopasvane 
| 445 | 198025 88121125 21 095023 76346067 gu Sate i 
| 446 | 198916 | 88716536 21.1187121 76403213 rinserkts fi 
; 447 | 199809 | 89314623 | 21.1423745 | 76460272 “002287136 | 
418 | 200v0e | sogis392 | 21.1660105 | 7.GorTed? “DoRDae LA | 
449 | 201601 | 90518819 | 21.1896201 | 716574133 |  ‘oo22RQ7171 
| | 450 | 202500 | 91125000 | 21.2132034 | 7.6630943 | 229295 
| 4 | Sesto, | grass: | St cbaeve0s | Feosrees “0B21 7208 
452 | 20130 | gesiodng | 212602016 | 6741303 | 002212389 i 
53 | 205209 | 92059677 | 21.28387967 | 7.s800857 | — 1002207506 i 
454 | 206116 | 93576664 S 3072758 | ressraes |. looBa0ao4s 
| 455 | 207025 | 94196375 | 21.8807200 |  7.6013717 “003197802 ii 
456 | 207936 94818816 21 3541565 76970023 “002109083 | 
457 | 208849 | 95443003 | 21.8775583 | 7. 7026246 “02188184 \ 
458 | 209764 | 96071912 | 21.4009316 7. @082388 “00218306 | 
| 459 | 210681 | 96702579 | 21.4242853 | 7.713848 002178649 " 
460 | 211600 | 9733600 21.4476 77194426 021734 ) 
te. | Sige | gyoreisy | aiaveoios | fracas |. <oopieig? 
462 | 213444 | 98611128 |  21.4941853 irposiat | “002164502 
| ~ 463 | 214369 | go2see7 | 21 5174348 | 7. 7361877 “02150827 | 
| 464 | 2t506 | goggrada | 1.540592 | 77417532 | [002 eae | 
465 | oieoas | 10nsi462s | 2icsesessy | ‘tira7siog | oozi50588 | 
466 | 217156 | 101194696 21.5870331 |  7.7528606 “002115923 
467 | 218089 | 101847563 21.6101828 | 7. 7584023 “pooidieer 
468 | 219024 | 102503232 | 21.6333077 | 7.760361 | “OO3136752 
$ p : | ; Ate as aoe A IIo ~UURLO0 (08 
469 | 219961 | 103161709 | 21.6564073 | 7. 7604620 “002132196 
470 | 220900 | 103823000 |  21.679483 7.77498 212766 
| Spies | qouernit | sicroessie | f-yeodood “posiaa142 | 
va | Seared | 105154048 | 217255610 | 7.759928 “D021 S64 | 
| 473 | aeare9 | t0sea3817 | 21.7485632 | 7.7914875 “DOR114165 
| aca | geaeve | 105496424 | oucevisaii | 7969745 “02100705 | 
4v5 | 905605 | 107iTI875 | 21 vo4i947 | 7.024538 “002105263 | 
476 | 206576 | 107850176 | 21 8174242 | Y_80r9e54 ‘002100810 
| 477 4 927529 | 108531333 |  21.8403207 |  7.8133892 002096436 [ 
| 473 | 228484 | 109215352 | 21 86B2L1 78188456 “02002080 
lt oor ODE Qe NBR f 5616 Te OnOsbee 
| 479 | 220441 | 109002289 "| 21.8860686 | 7.8242942 ‘002087683 
| 480 | 230400 | 110592000 |  21.9089023 | 7.820735: 208833: 
fa | Seat | itiseiet | sisirise | 4 8351068 “(02070002 
| 482 | 239304 | 111980168 |  21.9544984 #8403049 “002074680 
| 483 | 233989 | 1izvens7 | 21 9772610 | 718460134 “902070303 
| 484 | 231255 | 113379904 | 92/0000000 |  7l8514e44 “002066116 
485. | 235295 | 114081125 | 2210227155 | 78568281 “002061856 
| 486 | 2361968 | 114791256 | 22.0454077 | 7 8622242 “002057613 
487 | 237169 | 115501303 |  22.0680765 | — 7.8676130 “002058388 
ty | past | iioatiere | 22.000 | rsr2004 | “ooe0dgTs0 
43) | 239121 | 116930169 | 221133444 | 7 8783084 "002041999 
490 | 240100 | 117649000 |  22.1350436 | 7.8837352 | — .002040816 
tr | sires, | qissvorrt. | gecdsesies | 7.ss90016 | [902036660 
42 | 242064 | 119005188 | a2.1810730 | F180L4463 | “002032520 
493 | 243049 | 119823157 |  22.2036033 | esogzg17 | 109202838 
494 | 244036 | 120553784 | 22.2261108 |  7.9051204 | “902021201 
495 | 245025 | 121997375 | 2.248505 |  7_910450% “003020203 
| re Wiper > | sect io) 22 2485955 7. 9104599 002020202 
| 495 | e4oo16 | Teepesea6 | Re.zrIOsiD | T.91DTERR ‘oozsisize — | 
| 


9 
322 


Or 


| 
| 
| 
| 
| 


Squares. Cubes. 
217009 12276347 
248004 123505992 
249001 | 124251499 
250000 125000000 
251001 125751501 
25204 126506008 
253009 127263527 
254016 128024064 
255025 128787625 
256036 129554216 
257049 1303238843 
258064 1381096512 
259081 131872229 
260100 132651000 
261121 183482831 
262144 134217728 
263169 185005697 
264196 135796744 
265225 136590875 
266256 137388096 
267289 188188413 
268324 188991832 
269361 ° 139798359 
270400 140608000 
271441 141420761 
272484 142236648 
273529 143055667 
274576 143877824 
275625 144703125 
276676 145581576 
277729 146363183 
278784 147197952 
279841 148035889 
280900 148877000 
281961 149721291 
283024. 150568768 
284089 151419437 
285156 152273304 
286225 1531380375 
287296 153990656 
288369 154854153 
289444 155720872 
290521 156590819 
291600 157464000 
292681 158340421 
293764 159220088 
294849 160103007 
295936 160989184 
297025 161878625 
298116 162771336 
299209 163667323 
300304 164566592 
301401 165469149 
802500 166375000 
303601 167284151 
804704 168196603 
805809 169112377 
3806916 170031464 
808025 1709538875 
809136 171879616 
3810249 172298693, 

173741112 


311364 


TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS, 


Square 
Roots, 


222934968 
22.3159136 
22. 3383079 
223606798 
223830293 
22. 4053565 
22. 4276615 
22. 4499443 
22.4722051 
22. 4944438 
225166605 

25388553 
225610283 
225831796 
22. 6053091 
226274170 

2. 6495033 

2. 6715681 
22. 6936114 
227156334 
227376340 
23.7506134 
22. 7815715 
228035085 

2. 8254244 
22.8473193 
22. 8691933 
228910463 
229128785 
229346899 
229564806 
22. 782506 
23. 0000000 


28 0217289 
23 . 0434372 
23 .0651252 
23 . 0867923 
23. 1084400 
23. 1300670 
23 .1516788 
23 .17382605 
23 1948270 
23.2168735 
23.23879001 
23. 2594067 
23 .2808935 
23 38023604 
23 . 3238076 
23 .8452351 
23 8666429 
23 .3880311 
23 .4093998 
23 .4807490 


23. 4520788 
23.4733892 
23.4946802 
23 5159520 
23 5372046 
23 .5584380 
23 5796522 
23.6008474 
23. 6220236 


BOD OMMDMMDODOWDW WHWMOMWOOMWHMM-s-F aaa asa eda sett 


© cp co 


9210994 
. 9264085 
9317104 
9370053 
9422931 
9475739 
9528477 
9581144 
96383743 
9686271 
9738731 
9791122 
9843444 


. 9895697 
. 9947883 
0000000 
0052049 
.0104032 
.0155946 
0207794 
20259574 
.0811287 
0362935 
.0414515 
.0466030 
0517479 
0568862 
.0620180 
.0671482 
0722620 
0773743 
.0824800 
0875794 
.0926728 
.097'7589 
. 1028390 
.1079128 
1129803 
1180414 
1230962 
1281447 
1331870 
1882280 


1482529 
1452765 
1532939 
1583051 
1633102 
1683092 
1733020 
1782888 
1832695 
1882441 
1932127 
1981752 
.20313819 


8. 2080825 


8.21380271 
8.2179657 
8 .2228985 
8. 2278254 
8. 2327465 


323 


Reciprocals. 


| Cube Roots. 


002012072 
:002008032 
. 002004003 


002000000 
001996098 
.001992032 
.001988072 
001984127 
.001980198 
.001976255 
.001972387 
.001968504 
.001964637 


001960784 
.001956947 
.001953125 
001949318 
.001945525 
.001941748 
001937984 
. 601934236 - 
.001930502 
. 001926782 
001923077 
.001919386 
.001915709 
.001912046 
.001908397 
.001904762 
.001907141 
.001897533 
.001893939 
.001890359 
.001886792 
. 001883239 
.001879699 
.001876173 
001872659 
.001869159 
001865672 
.001862197 
.001858736 
. 001855288 
001851852 
001848429 
.001845018 
.001841621 
001838235 
.001834862 
.001831502 
.001828154 
.001824818 
.001821494 


001818182 
.001814882 
.001811594 
.001808318 
- 001805054 
.001801802 
.001798561 
001795832 
.001792115 


CUBE ROOTS, AND RECIPROCALS. 


No. | Squares.| Cubes. star oi 
559 | 312481 74676879 |  23.6431808 
560 313600 175616000 23 6643191 
561 314721 176558481 23 6854386 
562 315844 177504328 23 7065392 
563 316969 178453547 23 7276210 
564 318096 179406144 23. 7486842 
565 319225 180362125 23 7697286 
566 320356 181321496 23 7907545 
567 321489 182284263 23 .8117618 
568 | 322624 183250432 23 8327506 
569 323761 184220000 25 .8537209 
BYE 324900 185193000 23. 8746728 
571. | 326041 186169411 23 8956063 
572 327184 187149248 23,.9165215 
Aye 328329 188132517 239374184 
57 329476 189119224 23.9582971 
575 | 330625 190109375 23 9791576 
| 576 331776 191102976 24 0000000 
| 577 332929 192100033 24 0208243 
57 334084 193100552 24 0416306 
| 57 335241 194104529 24 0624188 
580 336400 195112000 24 0831891 
581 337561 196122941 24 1039416 
582 338724 197137368 24 1246762 
583 339889 198155287 24. 1453929 
584 341056 199176704 |  24.1660919 
585 342225 200201625 | 24.1867'732 
| 586 343396 201230056 24 2074369 
| 587 344569 202262003 24 2280829 
588 3457 903297472 | 24.2487113 
589 346921 204336469 24 2698222 
590 348100 205379000 24 2899156 
591 349281 206425071 | — 24.8104916 
592 350464 207474688 | 24.3310501 
593 351649 208527857 | 24.38515913 
594 352836 209584584 | 24.8721152 
595 354025 210644875 |  24.3926218 
596 355216 211708736 24 4131112 
597 356409 212776173 24 4335834 
598 357604 213847192 24 4540285 
599 358801 214921799 24 4744765 
600 360000 216000000 244948074 
601 361201 217081801 245153013 
602 | 362404 218167208 24 5356883 
| 603 | 363609 219256227 24 5560583 
604 364816 220348864 245764115 
| 605 366025 221445125 245967478 
606 | 367236 222545016 | 24.6170673 
607 368449 223648543 24 6373.00 
| 608 369664 294755712 246576560 
: 609 370881 295866529 24 6779254 
610 372100 226981000 24 .6981781 
G11 373321 228099131 24 7184142 
| 612 37444 229220928 | 24.7386338 
| | 613 375769 230346397 | 24.7588368 
| 614 76996 231475544 | 24.7790234 
| 615 378225 232608375 | 24 7991935 
616 | 3879456 933744896 | 24.8193473 
617 380689 934885113 |  24.8394847 
618 381924 236029032 24 8596058 
619 383161 237176659 24.8797106 
620 384400 238328000 248997992 


cots 
324 


Bsace" 
} 
| Cube Roots. | Reciprocals. | 
/ 
| §,2376614 001788909 
8.242506 001785714 
82474740 001782531 
8. 2523715 001779359 
| 8. 2572633 .001776199 
| §.2621492 001773050 
8.2670294 + .001769912 | 
82719039 001766784 
82767726 001763668 
82816355 001760563 
82864928 001757469 
8.2913444 001754386 
| 8.2961903 001751313 
| 8.2010804 001748252 
8.8058651 001745201 
8.3106941 001742160 
83155175 001739130 
8 3203353 001736111 
8.82514"5 .001733102 
8 8299542 001730104 
8.8847553 001727116 
8.3895509 001724138 
83443410 001721170 
8.3491256 001718213 
8.2539047 001715266 
8.586784 .001712329 
| 8.3624466 001709402 
83682095 001706485 
8.8729668 001708578 
8.3777188 001700680 
8. 2824653 .001697793 
|. 8.3872065 001694915 
83919423 .001692047 
88966729 .001689189 
8.4013981 .001686341 
8.4061180 001688502 
8. 4108326 001680672 
8.4155419 |. .001677852 
8. 4202460 .001675042 
8.4249448 001672241 
8. 4296383 001669449 
8 4343267 .001666667 
8.4390098 001663894 
8. 4436877 .001661130 
84483605 001658375 
84530281 . 001655629 
8. 4576906 001652893 
8.462347 .001650165 
8. 4670601 001647446 
84716471 001644737 
8. 4762892 .001642036 
8 4809261 .001639344 
84855579 -001636661 
8. 4901848 001633987 
8.4948065 001631321 
8 4994233 .601628664 
85040350 . 001626016 
8. 5086417 001623377 
85132435 001620746 
8.5178403 00161812: 
85224321 001615509 
8.5270189 .0C01612908 


TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS, 


Squares, | 


3885641 
886384 
388129 
3889376 
890625 
891876 
393129 
394384 
395641 
896900 
398161 
399424 
40689 
401956 
403225 
404496 
405769 
407044 
408321 
409600 
410881 
412164 
413449 
414736 
416025 
417316 
418609 
419904 
421201 
422500 
423801 
425104 
426409 
427716 
429025 
430336 
431649 
432964 
43,4281 
435600 
436921 
438244. 
439569 
440896 
442225 
443556 
444889 
446224 
447561 

448900 
450241 

451584 
452929 
454276 
455625 

456976 

458329 
459684 

461041 
462400 
463761 

465124 


ade 

Square | Cube Roots. | Reciprocals. 

hoots. 

24.9198716 8.5316009 .001610306 
24.9399278 8.53861780 -OOL607717 
24 9599679 8.5407501 .001605136 
24.9799920 | 8.5453173 .001602564 
25 . 0000000 8.5498797 .001600000 
25 .0199920 8.5544872 .001597444 
25 .0399681 8.5589899 .001594896 
25 0599282 8.56385377 .001592357 
25. 0798724 8. 5680807 . 001589825 
85 .0998008 8.5726189 .001587302 
25 1197134 85771523 . 001584786 
95 1396102 8.5816809 .001582278 
25. 1594913 8.5862047 001579779 
25 .1798566 8 .5907238 001577287 
25 .1992063 8.5952380 .001574803 
25.2190404 |  8.5997476 001572327 
25. 2388589 — | 8. 6042525 .001569859 
25 2586619 8. 6087526 001567398 
25.2784493 | 8.6132480 001564945 
25.29822138. | 8.6177388 .001562500 
25.38179778 |  8.6222248 001560062 
25 .33877189 | 8.6267063 . 001557632 
25.3574447 | © §.63118380 .001555210 
25.38771551 | °8.6356551 .001552795 
25 38968502 | 8 .6401226 .001550388 
25.4165301 8.6445855 .001547988 
25 .4861947 |  8.6490437 .001545595 
25.4558441 | 8.65384974 .001543210 
25.4754784 | 8.6579465 .001540832 
25.4950976 | 8.6623911 .001538462 
25.5147016 8. 6668310 .001536098 
25 .5842907 8.6712665 001533742 
25 .5588647 | 8.6756974 -001531894 
20D (o42a7 | 8.6801237 -001529052 
25.5929678 | 8.6845456 -001526718 
25.6124969 | 8.6889630 001524390 
25 .6320112 8.6933759 .001522070 
25..6515107 8.6977843 .001519757 
25.6709953 |  8.7021882 .001517451 
25.6904652 | 8.7065877 .001515152 
25.7099203 |  8.7109827 - 001512859 
25.7298607 | 8.7158734 -001510574 
25 . 7487864. 8.7197596 -001508296 
25 .7681975 8.7241414 -001506024 
25 ..7875939 8. 7285187 . 001503759 
25. 8069758 8.7328918 .001501502 
25.8263431 8.7372604 .001499250 
25.8456960 | 8.7416246 001497006 
25 .8650343 | 8 .7459846 .001494768 
25 . 8848582 8.503401 .001492537 
25 9036677 8.546913 .001490313 
25 . 9229628 8 .7590883 .001488095 
25 . 9422485 §.7638809 001485884 
25.9615100 8.7677192 001483680 
25 . 9807621 8.720532 .001481481 
26 .0000000 8.7'763830 .001479290 
26 0192237 8. 7807084 .001477105 
26.0384331 8.'7850296 001474926 
25 0576284 8.7893466 .001472754 
26 0768096 8.7936593 .001470588 
26 .0959767 8.7979679 001468429 
26 .1151297 8 .8022721 .001466276 


90K 
3825 


CUBE ROOTS, 


| 
| 


AND RECIPROCALS. 


| 


No. eauareey Cubes. | ee: | Cube Roots, | Reciprocals. 
683 | 466489 318611987 26 . 1342687 8.8065722 001464129 
681 | 467856 320013504 | 26.1588937 8. 8108681 001461988 
685 | 469225 321419125 | 26.1725047 8.8151598 001459854 
686 470596 322828856 | 26.1916017 8.8194474 001457726 
687 | 471969 824242703 26 2106848 8. 8237307 001455604 
G88 | 473344 325660672 26 2297541 8. 8280099 001453488 
689 | 474721 327082769 26 2488095 8 .8322850 001451379 
690 | 476100 998509000 | 26.2678511 |  8.8865559 001449275 
691 477481 329939371 | 26.2868789 | 8. 8408227 001447178 
692 478854 331373888 |  26.3058929 88450854 001445087 
693 | 480249 | 332812557 | 263248932 | 8.8493440 001443001 
694 | 481636 | 334255384 | 26.8488797 | 8.8535985 | 001440922 
695 483025 335702375 | 26.8628527 | 8.8578489 | 001438849 
696 481416 | 387152536 | 26.3818119 | 8 .8620952 001436782 
697 | 485809 | 338608873 | 26 4007576 8.8663375 001434720 
698 | 487204 | 340068392 | — 26.4196896 3.8705757 =|, .001482665 
699 | 488601 | 341532099 | 26 4886081 8.8748099 | .001480615 
700 | 490000 | 343000000 | 26 4575131 8.8790400 |  .001428571 
VOL | 491401 344472101 | 26.4764046 | —8.8832661 001426534 
702 | 492804 345048408 | 26.4952826 | 8.8874882 | 001424501 
"03 | 494209 | 347428927 965141472 | 8.8917063 | = 001422475 
704 | 495616 | 3848918664 265329983 | 8.8959204 |  .001420455 
705 497025 | 350402625 | 26.5518361 | 8.9001304 |  .001418440 
706 498436 | 351895816 |  26.5706605 89043366 | 001416431 
707 499849 353393243 |  26.5894716 3.9085387 | .001414427 
708 | 501264 | 354894912 | 26.6082694 8.9127369 001412429 
709 | 502681 356400829 | 26 6270539 | 8.9169311 001410437 
"10 | 504100 357911000 |  26.6458252 8.9211214 , 001408451 
711 505521 | 359425431 26 6645833 | 8.9253078 | 001406470 
712 | 506944 | 360944128 26 6833281 8.9294902. | 001404494 
#13 | 508369 | 362467097 26. 7020598 8.9336687 001402525 
ee 509796 863994344 26. 72077 89378433 | 001400560 
15 | 511225 365525875 26 .7394839 8.9420140 | .001398601 
716 512656 367061696 26 . 7581763 8.9461809 001396648 
717 514089 368601813 26 .7768557 8.9503438 | 001394700 
W718 .| 515524 3701462382 26 .7 955220 89545029 | .001892758 
519 | 516961 371694959 268141754 8.9586581 001390821 
"20 | 518400 373248000 |  26.8328157 8. 9628095 001388889 
m21 | 519841 374805361 |  26.8514432 8. 9669570 401886963 
(22 521284 | 876367048 26 8700570 8.9711007 001885042 
723 522729 377933067 |  26.8886593 8. 9752406 .001383126 
724 | 524176 379503424 26 . 9072481 89793766 .001881215 
125 525625 381078125 |  26.9258240 8 .9835089 001379310 
726 527076 382657176 26 . 9443872 8.9876373 .001377410 
1270 528529 | 384240583 26 . 9629375 8. 9917620 .001375516 
728 529984. | 385828352 26 .9814751 8 . 9958829 001873626 
729 531441 387420489 27 .0000000 9 .0000000 001371742 
730 532900 389017000 27 0185122 90041184 001369863 
731 | 534361 390617891 27 0370117 90082229 | 001367989 
732 | 535824 392222168 27 0554985 9 0128288 .001366120 
733 | 587289 393832837 27 07389727 9 .0164309 001364256 
734 | 538756 395446904 27 .0924344 9 0205293 001362898 
"735 | 640225 397065375 | 27.1108834 9 0246239 .001860544 
726 | 541696 398688256 | 27.1293199 9 0287149 001358696 
737 | 543169 400315553 27 1477439 9 0328021 | .001356852 
mga | 544644 | 401947272 | 27.1661554 9 0368857 |  .001855014 
nag | 546121 | 403583419 |  27.1845544 90409655 |  .001353180 
"40 | 547600 | 405224000 | 97.2029410 | 9.0450419 .001351351 
TAL 549081 406869021 | 27.2218152. | 9.0491142 001849528 
742 550564 408518488 | 27.23896769 | 9.0531831 | 001347709 
743 552049 | 410172407 | 27. 2580263 90572482 | .001845895 
744 553536 | 411830784 | 27.2763634 9,0613098 {|  .001844086 


326 


=e 


hI 
i 
| 
} 


Squares. 


TC) 


Oo 
— 


e 


~~ ~> =] ~t =} ~ 
OTOL UL OT Or 


“2 Oo OUHS Co 0 


OV oror 
Laie 2) 


J 3-3 5 


~ 


> SS 
i=) 


J 3 3-5 5 
VNININS 
et 


OU CO a 


2-2 +) 
COIS 


i ae ee 


555025 
556516 
558009 
559504 
561001 
562500 
564001 
565504 
567009 
568516 
570025 
571586 
573049 
574564 
576081 
577600 
579121 
580644 
582169 
583696 
5852 
586756 
588289 
589824 


591361 


592900 
594441 
595984 
597529 
599076 
600625 
602176 
603729 
605284 
606841 
608400 
609961 
611524 
613089 
614656 
616225 
617796 
619369 
620944 
62252 


I 


624100 
625681 
627264 
628849 
630436 
632025 
633616 
635209 
636804. 

338401 
640000 
641601 
648204 
644809 
646416 
648025 
649636 


Cubes. 


413493625 
415160936 
416832723 
418508992 
420189749 
421875000 
423564751 
425259008 
426957777 
428661064 
430368875 
432081216 
433798093 
435519512 
437245476 

438976000 
440711081 
442450728 
444194947 
445943744 
447697125 
449455096 
451217663 
452984832 
454756609 
456533000 
458314011 
460099648 
461889917 
463684824 
465484375 
467288576 
469097433 
470910952 
472729139 


474552000 
476379541 
478211768 
480048687 
481890304. 
483736625 
485587656 
487443403 
489303872 
491169069 


493039000 
494913671 
496793088 
498677257 
500566184 
502459875 
504358336 
506261573 
508169592 
510082399 


512000000 
513922401 
515849608 
517781627 
519718464 
521660125 
523606616 


TABLE XXTII.—SQUARES, CUBES, SQUARE ROOTS, 


7 
7.3678644 
7 
7 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


27 2946881 
273180006 
7 .8813007 


"8495887 


38861279 
"4043792 


27. 4226184 
27. 4408455 
274590604 
27 47726828 
27 .4954542 
(51863830 
% 5917998 
27 5499546 
27 5680975 
275862284 
27 60438475 
27.62.4546 
7 6405499 


", 6586334 


276767050 
27. 6947648 
27 .7'128129 


", 7808492 


27. 7488739 
27. 7668868 
27.7848880 
27 8028775 
27 8208555 


8388218 


27 8567766 
«8747197 


’, 8926514 


279105715 
27 9284801 
27 . 9463772 
9642629 
', 9821372 
8. 0000000 
8.0178515 
28.0356915 
8 .0535203 
280718377 
28. 0891438 


. 1069886 


28 . 1247222 
3. 1424946 
8. 1602557 
28. 1780056 
28.1957444 
28 2134720 
28.2311884 
28 .2488938 
28 .2665881 


2842712 


28 3019434 


9196045 
3372546 
.8048938 


28.3725219 


.3901391 


9.0653677 
9 .0694220 
9.0734726 
9.0775197 
9.0815631 
9 .0856030 
9. 0896392 
9.0936719 
90977010 
9.1017265 
9.1057485 
9.1097669 
9.1187818 
9.1177931 
9.1218010 


9.1258053 
9.1298061 
9.1388034 
9.137°7971 
9.141787 

9.145742 
9.1497576 
9.15387375 
9.157'7139 
9.1616869 
9.1656565 
9. 1666225 
91785852 
9.1775445 
9.181E003 
9. 1854527 
9.1894018 
91988474 
9.1972897 
9.201226 


9.2051641 
9 2090962 
9 2180250 
9.2169505 
9.2208726 
92247914 
9 .2287068 
9.2526189 
9.286527 
92404383 
9. 2443355 
9. 2462344 
9. 2521300 
9 .2560224 
9 .2599114 
9.2637973 
9 .2676798 
9.2715592 
9.2754352 
9.2798081 


9. 2881777 
9 2870440 
9.2909072 
9.2947671 
9 2986239 
9 8024775 
9 .3063278 


327 


.001342282 
.0013840483 
.00133886&8 
.-001886898 
.001835113 
. 001333333 
-0U13881558 
-001829787 
. 0013828021 
- 001826260 
. 001824503 
.001322751 
. 001821004 
-001319261 
.C01317523 


.001815789 
.C01314060 
-C01812336 
-001310616 
-001808901 
.001307190 
.001805483 
.001803781 
.0013802083 
- 001300390 


001298701 
.001297017 
001295337 
-001293661 
.001291990 
001280323 
.001288660 
.001287001 
001285347 
.001283697 
001282051 
. 001280410 
.001278772 
001277139 
001275510 
001273885 
001272265 
.001270648 
.C01269036 
001267427 


001265823 
.C01264223 
001262626 
.001261034 
001259446 
.001257862 
001256281 
.001254705 
.001258133 
. 001251564 


001250000 
.001248489 
.001246883 
001245330 
.001243781 
001242236 
001240695 | 


CUBE ROOTS, AND RECIPROCALS. 


\ | Cc 7Tra 
No. | Squares. Cubes. Ph otan 
807 651249 525557943 28 .4077454 
808 652864 527514112 28 . 4258408 
809 | 654481 529475129 28 .4429253 
810 656100 531441000 28 .4604989 
811 657721 5383411731 28 .4780617 
812 659344 535387328 28 .4956137 
813 660969 537867797 28.5131549 
814 662596 589353144 28 .5306852 
815 654225 541343375 23.5482048 
816 665856 543338496 28 .5657137 
817 | 667489 545338513 28 .58382119 
818 | 669124 547343432 28 . 6006995 
819 670761 5493538259 28,6181760 
82 672400 551368000 28 .635642 
821. | 674041 553387661 286530976 
| 822 675684 555412248 28 6705424 
82% 677329 557441767 288 6879766 
| 82 678976 55947622 28 .'7054002 
| 825 680625 561515625 28.72281382 
826 682276 563559976 287402157 
| 827 683929 565609288 | 28.7576077 
) 828 685584 567663552 28.7749891 
| 829 687241 569722789 >| 23.'7923601 
830 688900 571787000 28 .8097206 
831 690561 573856191 288270706 
832 692224 575930368 28 .8444102 
: 833 693889 578009537 28 .8617394 
) 834 695556 580093704. 28.8790582 
: 835 697225 582182875 28.8963666 
: 836 698896 584277056 28 .9186646 
837’ | 700569 586376253 28 9309523 
838 702244 588480472 28 . 9482297 
839 703921 590589719 28.9654967 
840 705600 592704000 28 .9827535 
841 707281 594823321 29 . 0000000 
842 708964 596947688 29.0172363 
843 | 710649 599077107 29. 034462: 
844, | 712386 601211584 29.0516781 
845 714025 603351125 29. 0688837 
846 | 715716 605495736 29 .0860791 
347 | =. 717409 607645423 29 .1032644. 
848 719104 609800192 29 . 1204396 
849 720801 611960049 29.1376046 
850 722500 614125000 29.1547595 
351 | . 724201 616295051 29.1719043 
852 72590 618470208 29.1890390 
853 727609 620650477 29.2061637 
854 | 729316 22835864. 29 2232784 
| 855 731025 625026375 29 .2403830 
| 856 732736 627222016 29. 2574777 
857 734449 629422793 29.2745623 
| 858 736164 631628712 29 .291687 
| rola\ Pima ies UE ifoteil 6338397769 29.38087018 
860 739600 636056000 29 3257566 
861 741321 638277381 29.3428015 
862 743044 640503928 29 .38598365 
863 744769 642735647 29.3768616 
864 | 746496 644972544 29 3938769 
865 | | 748225 647214625 29 4108823 
: 866 | . 749956 649461896 294278779 
: 867. | | 751689 651714363 29 4448637 
868 | | 7538424 653972032 29 .4618397 


3 


28 


Cube Roots. 


| Reciprocals. 


.8101750 
.8140190 


.B178599 


e 304 
. 38381916 
.38370167 
. 38408386 
.38446575 
38484731 
8522857 
.0- 60952 
.8599016 
.8537049 
. 8675051 


9 .38718022 
9.3750963 


3788873 


9. 38826752 
9 .3864600 
9 .3902419 
9. 3940206 


8977964 


9 
9.4015691 
") 


4058387 


9 .4091054 
9 .4128690 


Ow 


SODOOO OOsewssd 


la) 


.4166297 
.4203873 
.4241420 
.4278936 
.4816423 
.4358880 
.43913807 
.4428704. 
.4466072 
.4503410 
.4540719 
.4577999 
.4615249 
.4652470 
.4689661 
4726824 
.4763957 
.4801061 
.4838136 
.4875182 
.4912200 
.4949188 
.4986147 
.5023078 
_ 5059980 


5096854 
.5138699 
.5170515 
. 5207308 
5244063 
5280794. 
9817497 
5354172 
.5390818 


.001289157 
.00123762 

.001286094 
.001234568 , 
.001283046 
.001231527 
.001230012 
.001228501 
. 001226994. 
. 001225490 
.001223990 
.001222494 
- 001221001 


.001219512 
.001218027 
. 001216545 
.001215067 
.001218592 
.001212121 
.001210654 
.001209190 
.001207729 
.001206273 
.001204819 
.001203369 
.001201923 
.001200480 
.001199041 
001197605 
.001196172 
.001194743 
.001198317 
.001191895 


.001190476 
.001189061 
.001187648 
001186240 
.001184834 
.001183432 
.001182033 
.001180638 
001179245 
.001177856 
001176471 
.001175088 
.001173709 
.001172833 
.001170960 
.001169591 
.001166224 
.001166861 
.001165501 
; .001164144 


.001162791 
.001161440 
.001160093 
.001158749 
.001157407 
.001156069 
.001154734 
.001158403 
, 001152074 


TABLE XXIII.—SQUARES, CUBES, 


SQUARE ROOTS. 


| 


80.4959014 


ae 


| No. Squares. Cubes. Tks | Cube Roots. Reciprocals. 
a : = 
869 755161 656234909 | 29.4788059 9 5427437 001150748 
870 756900 658503000 | 294957624 9.5464027 | 001149425 
aff 758641 660776311 | 29.5127091 9.5500589 | .001148106 | 
872 760384 663054848 | 29.5296461 | = 9.5537123 | _ .001146789 
873 762129 665338617 | 29.5465784 | 9.5573630 001145475 
87 768876 667627624 | 29.5634910 | 9.5610108 | .001144165 
875 765625 669921875 | 29.5803989 |  9.5646559 001142857 
87 767376 6722213876 | 29.5972972 1} 9.5682982 | .001141553 
877 769129 6745261338 | 29.6141858 9.57193877 | .001140251 
78 770884 | 676836152 | 29.6310648 9.5755745 | 001188952 
879 772641 79151439 | 29.6479342 9.579208 | .001137656 
880 774400 | 681472000 | 29.66479389 | 9.5828397 | .001136364 
Sot a  COLOk 683797841 | 29.6816442 | 9.5864682 | .001135074 
882 777924 686128968 | 29.6984848 | 9.5900939 | .001133787 
883 | 779689 688465387 | 29.7153159 | 9.59387169 | .001182503 
884 | 781456 690807104 | 29.7321875 | 9.5973873 | 001131222 
a 885 | 783225 693154125 | 29.7489496 | 9.6009548 | .001129944 
Hi 886 784996 | 695506456 | 29.7657521 | 9.6045696 | 001128668 
Hil: 887 786769 697864103 | 29.7825452 | 9.6081817 | .001127896 
Heat, 888 | 788544 700227072 =| 29.7998289 | 9.6117911 | .001126126 
Ma 889 | 7903821 | 702595369 / 29.81610380 | 9.6153977 | .001124859 
Wie 890 | 792100 | 704969000 | 29.8828678 | 9.6190017 | .001123596 
891 | 793881 | TO7347971 | 29.8496281 | 9.6226080 | .001122334 
i 892 795664 709732288 | 29.8663690 | 9.6262016 | .001121076 
ist 893 797449 712121957 | 29.8831056 9.62979%5 001119821 
i 894 799236 714516984 | 29.8998328 | 9.6338907 .001118568 
ih 895 801025 716917375 | 29.9165506 | 9.6369812 .001117318 
Hi 896 802816 719323186 | 29.9882591 | 9.6405690 .001116071 
897 804609 721734278 | 29.94995838 | 9.6441542 .001114827 
898 806404 724150792 | 29.9666481 9.647367 .001113586 
899 808201 726572699 = -29.9838287 | 9.6513166 .001112847 
| | 

900 810000 729000000 | 30.0000000 | 9.6548938 .001111111 
901 811801 | 731482701 | 30.0166620 | 9.6584684 .001109878 
902 813604 7188870808 30.0383148 | 9.6620403 .001108647 
903 815409 736314327 80 .0499584 9. 6656096 .001107420 
904. 817216 788763264 30 .0665928 96691762 .001106195 
905 819025 741217625 30.08382179 9 .6727403 001104972 
906 820836 743677416 | 30.0998339 9.67638017 .001103753 
907 822649 746142643 | 30.1164407 9.6798604 .001102536 
908 824464 748613312 | 30.1330383 9 .6834166 .001101322 
909 826281 751089429 | 30.1496269 9 .6869701 .001100110 
910 828100 753571000 | 30. 1662063 9.6905211 .001098901 
911 829921 756058031 80. 1827765 9.6940694 .001097695 
912 831744 758550528 | 30.199337'7 9.6976151 | .001096491 
all 913 833569 761048497 80.2158899 9.011583 .001095290 
914 835396 | 763551944 380 . 2824329 9. 7046989 .001094092 
ait 915 837225 766060875 380. 2489669 9 7082869 .001092896 
4 916 839056 | 768575296 30.2654919 9.7117728 .001091703 
917 840889 | 771095213 30. 2820079 9.7158051 .001090513 
918 842724 | 7773620632 80 .2985148 9 7188354 .001089325 
919 844561 | 776151559 80.3150128 9 7223631 . 0010881389 

920 846400 | ee 30 .38315018 9 7258883 001086957 

921 | 848241 | 781229961 30.34'79818 9 .7294109 .001085776 
922 850084 | 783777448 80 .3644529 9 7329309 .001084599 
923 851929 | 786330467 80.3809151 9.7364484 | .001083423 
924 853776 | 788889024 80 .8978683 9.7399634 | .001082251 
925 855625 | 791453125 | 80.4138127 9 .'7434758 .001081081 
926 857476 | 794022776 | 30.4302481 97469857 .001079914 
| 927 859329 796597983 30 .4466747 9. 7504930 .001078749 
| 928 861184 T99178752 80 .4630924 9.7539979 .001077586 
929 863041 801765089 30.4795013 9.7575002 .001076426 
930 864900 804357000 9.7610001 .001075269 


CUBE ROOTS, AND RECIPROCALS. 


! 


| 


} | 
No. Squares, | Cubes. Pete | Cube Roots. | Reciprocals. 
931 866761 806954491 80.5122926 97644974 .001074114 
932 868624 | 809557568 30 .5286750 9.7679922 .001072961 
933 870489 | 812166287 | 30.5450487 9.7'714845 .001071811 
934 872356 814780504 | 380.5614136 9.749743 .001070664 
935 | 874225 817400375 30.5777697 9.7784616 .001069519 
936 | 876096 820025856 380.5941171 9.7819466 .001068576 
937 =| 877969 322656953 30.6104557 97854288 001067236 | 
9388. | 879844 | Sati 3672 80. 6267857 9.7889087 .001066098 | 
9389 881721 | 27936019 380.6481069 | 9.7928861 .001064663 
940 | 883600 830584000 30.6594194 | 9.7958611 . 0010688380 
941 | 885481 833257621 30.6757233 | 9.7993236 | .001062699 
942 | 887364 835896888 30.6920185 | ee | 001061571 
943 | 889249 | 838561807 | 380.7083051 | 9.8062711 .001060445 
944. |. 891136 | 841282384 40.7245880 | 9.80973862 .001059822 
945. | 898025 843908625 | 30.7408523 | 98131989 .001058201 
946 | 894916 846590536 30.7571130 | 9.8166591 .001057082 
947 | 896809 849278123 30.7788651 | 9.8201169 .001055966 
948 | 898704 851971392 30.7896086 | 9.8285723 001054852 
$49 | 900601 | 854670349 | 30.8058436 9 8270252 -001053741 
950 | 902500 | 857375000 30.8220700 | 9.8304757 | .001052632 
951 904401 | 860085851 30.8882879 | 9.838389288 .001051525 
952 906304 862801408 80.8544972 9.8378695 .001050420 
953 | = 908209 865523177 380.8706981 9. 8408127 .001049318 
954. | 910116 868250664 | 80.8868904 9 8442536 .001048218 
95 | 912025 | 870983875 30. 9030743 98476920 .001047120 
956 | 913936 873722816 80. 9192497 9.8511280 .001046025 
957 915849 876467493 30.9354166 | 9.8545617 | .001044932 
958 | 917764 879217912 30.9515 751- | = 9.8579929 .0010438841 
959 | 919681 881974079 | 380.9677251 , 9.8614218 | .001042753 
960 | 921600 | 884736000 80. 9828668 9.€648483 | .001041667 
961 | 928521 | 887508681 | 81.0000C00 9.68 2724 001040583 
962 925444 890277128 | 31.0161248 9.8716941 001039501 
963 927369 898056847 | 381.0822413 9 8751135 001088422 
964 929296 895841344 ; 81.0488494 9.6785805 | .001087344 
965 931225 898632125 | 31.0644491 9.819451 | .001086269 
966 933156 901428696 | 381.0805405 9. €85357 001085197 
967 985089 904231063 | 81.0966286 9 .E887673 .001084126 
968 937024 907039232 31. 1126984 9.8921749 .001083058 
969 938961 909858209 381. 1287648 9.8955€01 |  .001081992 
970 940900 912673000 | 381.1448230 9.8959830 | .001030928 
971 942841 915498611 381.1608729 9. C028&85 .001029866 
972 944784 918380048 81 .1769145 9. seth .001 028807 
973 946729 921167317 81 .1929479 9.909177 .001027'749 
97. 948676 924010424 | 81.2089731 | 99125419 001026694. 
975 950625 926859375 | 31.2249900 |! 9.9159624 . 001025641 
76 952576 929714176 | 381.2409967 {| 9.9198513 .001024590 
O77 | 954529 9825748838 | 81.2569992 | 9.9227379 .001023541 
978 956484 935441352 | 381.2729915 9 .9261222 001022495 
979 958441 938318739 | 31.2889757 | 9.9295042 .001021450 
980 960400 941192000 | 381.8049517 | 9.9828839 .C01020408 
981 962361 944076141 | 31.8209195 |; 9.9862613 .001019368 
982 964324 946966168 31 .3868792 | ~ 9.9896868 .001018330 
983 966289 949862087 31.3528308 | 9 9430092 .001017294 
984 968256 952763904 81.38687743 9 .94638797 .001016260 
985 970225 955671625 31.3847097 | 9.9497479 .001015228 
986 972196 958585256 | 81.4006369 | 9 .9581138 .001014199 
987 974169 961504803 81 .4165561 9.9564775 | .0010138171 
988 | 976144 964480272 31 .4324673 9 . 95983889 .001012146 
989 978121 967361669 31.4483 3704 | 9.9631981 .€01011122 
990 980100 970299000 81 .4642654 9 9665549, .001010101 
991 982081 973242271 81 .4801525 | 9.9699095 .001009082 
992 | 984064 976191488 381.4960315 | 9.9782619 . 001008065 | 
—-———~d 


380 


| 


i 1049 | 1100401 
Hit 1050 1102500 


1051 1104601 
1052 1106704 
1053 1108809 
1054 1110916 


1154320649 
1157625000 
1160935651 
1164252608 
1167575877 
1170905464 


No. Squares. Cubes. ee 
| 
| 

993 986049 979146657 | 31.5119025 
994 988036 982107784. 315277655 
995 990025 985074875 | 31.5436206 
996 992016 988047936 315594677 
997 994009 99102697: 31.5753068 
| = 998 996004 994011992 31.5911380 

999 998001 997002999 31 6069613 

1000 1000000 1000000000 | 31.6227766 

1001 1002001 1003003001 | 31.6385840 

1002 1004004 1006012008 | 31.6543836 

me 1003 1006009 1009027027 | 81.6701752 
Aan 1004 1008016 1012.\48064 | 31.6859590 
Be | 1005 1010025 1015075125 | 381.7017349 
1096 1012036 1018108216 | 81.7175030 

1007 1014049 1021147343 | 31.7332633 

: 1098 1016064 1024192312 | 31.7490157 
aah 1009 1018081 1027243729 ) 31.7647603 
ia 1010 1020100 103)301000 | 31.7804972 

HAA | 1011 1022121 1033361331 | 31.7962262 
iN 1012 1024144 1036433728 | 31,8119474 
He | 1013 1026169 1039509197 318276609 
Hil 1014 1028196 1042590744 : 31.8433666 
We ae 1015 1030225 1045678375 318590646 
Bee a 1016 1032256 1048772096 31.8747549 
ah ~ 4017 1034289 1051871913 31. 8904374 

Wit 1018 1036324 1054977832 319061123 

na 1019 1038361 1058089859 | 31.9217794 
Hi 1020 1040400 1061208000 | 31.9374388 
Ht 1021 1042441 1081332261 319530906 

all 1022 1044484 1067462648 31. 9687347 
in| 102 1046529 1070599167 | 31.9843712 
Bani 1021 1048576 1073741824 | 32.0000000 
| 1025 1050825 1076890625 32.0156212 

i} 1026 1052876 1030045576 32.0312348 
1027 1054729 1083206683 32.0468407 

102: 1056784 1036373952 | 82.0624391 

1029 1058841 1039547339 | 32.0780298 

1030 1080900 1092727009 | 82.0936131 

1031 1062961 1095912791 | 32.1091887 

1032 1085024 1099104768 32. 1217568 

1033 1067089 1102302937 32. 1403173 

1034 1089156 1105507304 32.1558704 

1035 1071225 1108717875 32.1714159 

1036 1073296 1111934656 | 32.1869539 

1037 1075369 1115157653 32. 2024844 

me | 1038 1077444 1118386872 | 382.2180074 
i 1039 1079521 1121622319 32 2335229 
il 1040 1081600 1124864000 322490310 
aul 1044 1083681 1128111921 322645316 
siti 1042 1085764 1131366088 32. 2800248 
1043 1087849 1134626507 32.2955105 

1044 1089936 1137893184 32.3109888 

1045 1092025 1141166125 32. 3264598 

HT 1046 1094116 1144445336 32. 3419233 
Wi 1047 1096209 1147730823 32.3573794 
it | 1048 1098304 1151022592 32. 3728281 


2. 3882695 


4037035 


.4191301 
4345495 
.4499615 
4653662 


TABLE XXIII.—_SQUARES, CUBES, ETC. 


| ee 
} | 


| { 


| Cube Roots. 'Reciprocals. 


| 
| | 
| | 


| 9.9766120 


.001007049 
| 9,9799599 .001006036 
99833055 | .001005025 
9.986488 _; .001004016 
9.9899900 | .001003009 

| 


9 .99338289 .001002004 
|  9,9966656 .001001001 
10.0000000 .001000000 
10.0033322 | .0009990010 
| 10.0066622 .0009980040 
10.0099899 .0009970090 
10.0133155 | .0009960159 


10.0166389 | .0009950249 
10.0199601 | .0009940358 
10.0232791 | .00099380487 
10.0265958° | 0009920635 


10.0299104 .0009910808 
10.0332228 0909900990 
10.0365330 0009891197 
10 .0398410 . 0009881423 
10.0431469 .0009871668 
10.0464506 | .0009861933 
10.0497521 .0009852217 
10.0580514 - 0009842520 
100563485 . 0009832842 
10.0596485 . 0009823183 
10.0629364 .0009813543 
10.066227 . 0009803922 


10.0695156 | .0009794319 
10. 0728020 0009784736 
10.0760863 .0009775171 
10.0793684 0009765625 
10.0826484 .0009756098 


10.0859262 .0009746589 
10.0892019 .0009737098 
10.0924755 .0009727626 
10.0957469 .0009718173 
10.0990163 0009708738 


10. 1022835 .0009699321 


10. 1055487 -00.9689922 
10.1088117 . 0009680542 
10. 1120726 .0009671180 


10. 1153314 . 0009661836 
10.1185882 .0009652510 
10. 1218428 .0009643202 
10. 1250953 .0009633911 
10. 1283457 -0009624639 
10.13815941 .0009615385 


10. 1348403 0009606148 


* 10.1880845 .0009596929 


10. 14138266 .0009587'738 
10. 1445667 .0009578544 
10.1478047 0009569378 
10.1510406 .0009560229 
10. 1542744 .0009551098 
10. 1575062 0009541 985 
10.1607359 .0009532888 
10. 1639636 .0009523810 
10.1671893 .0009514748 
10.1704129 .0009505703 
10.1736344 .0009496676 


10.1768539 .0009487666 


301 


—— 


| Wo. 100 L. 000.] 


[No. 109 L. 040. | 


TABLE XXTV.—LOGARITHMS OF 


NUMBERS. 


igo eee a ee | 2 3 4 5 6 7 | 8.94 9.) pitt 
aes | ns Se | | 
100 | 000000 0434 | 0868 1301 | 1734 || 2166 2598 3029 | 3461 | 3891 | 432 
1 4321 | 4751 | 5181 | 5609 | 60388 | 6466 | 6804 T7321 | 7748 | 8174 | 428 ! 
2 8600 9026 | 9451 9876 | | | : | 
= os | 0300 || 0724 | 1147 | 1570 | 1993 | 2415 | 424 
8 | 012887 | 3259 | 3680 | 4100 | 4521 || 4940 | 5360 | 8779 | 6197 | 6616 | 420 
4 7033 | 7451 | 7868 | 8284 | 8700 || 9116 | 9532 | 9947 | | 
| — ——}| 0361 | 0775 | 416 
5 | 021189 | 1603 | 2016 | 2428 | 2841 || 8252 | 3664 | 4075 | 4486 | 4896 | 412 
6 | 5306 | 5715 | 6125 | 6583 | 6942 || 7350 | 7757 | 8164 | 8571 | 8978 | 408 
7 | 9384 | 9789 | | | | | = 
0195 | 0600 | 1004 || 1408 | 1812 | 2216 | 2619 | 8021 | 404 
8 | 033424 | 3826 | 4227 | 4628 | 5029 || 5430 | 5830 | 6230 | 6629 | 7028 | .400 
9 | 7426'| 7825 | 8223 | 8620 | 9017 || 9414 | 9811 | | 
| 04 | | i | | o207 | 0602 | 0998 | 397 
PROPORTIONAL PARTS. 
ree | | | 
dy 2 els Ge Cane 2g a ON | 5 ify SM ea 7p 8 9 
| | | i | | 
434 | 43.4) &6.8 | 130.2 | 173.6 | 217.0 | 260.4 | 303.8 | 347.2 | 390.6 
433. | 43.3| 86.6.) 129.91 173.2} 216.5) 259.8 | 3803.1 | 346.4 | 389.7 
432 3.2| 86.4 | 129.6 | 172.8 | 216.0 | 259.2 | 302.4 | 345.6 | 388.8 
431 | 43.1} 86.2 | 129.3! 172.4% 215.5 | 258.6) 301.7 | 344.8 387.9 
43 3.0 | 86.0. | 129.0] 172.0 | 215.0 | 258.0 |) 301.0 | 344.0) 387.0 
4299 | 42.9} 85.8 28.71 171.6 }- 214.5) 257.4 | 300.3) 348.2% 386.1 
428 9 § |. 85.6 128.4 | 171.21) 214.0 | 256.8 299.6 | 342.4 | 385.2 
427 | 42.71 85. 128.1.|° 170.8 | 218.5 | 256.2) 298.9 | 841.6) 384.3 
426 | 42.6} 85.2 | 127.8| 170.4) 218.0! 2556); 293.2 | 340.8 | 883.4 
425 |142.5 | 85.0 127.5 | 170.0} 212.5 | 255.0) 297.5 | 340.0 | 382.5 | 
494 | 42.4) 84.8 127.2 | 169.6 | 212.0 | 254.4 | 296.8 | 389.2 | 381.6 
423 | 42.3 | 84.6 126.6 169.2 | 211.5 | 258.8! 296.1; 838.4 | 380.7 
92 | 42.2) 84.4 126.6 | 168.8 | 211.0°| 258.2} 295.4 | 337.6 | 379.8 
421. | 42.1) 84.2 126.3 | 168.4 | 210.5 | 252.6 | 294.7 | 336.8) 378.9 
420 | 42.0 | 84.0 126.0 | 168.0 | 210.0 | 252.0} 294.0 | 336.0 | 378.0 
419 | 41.9 | 83.8 125.7 | 167.6 | 209.5 | 251.4! 293.3 | 335.2 | 377.1 
418 | 41.8; 83.6 125.4 | 167.2} 209.0 | 250.8 | 292.6 | 334.4 | 376.2 
41V>| 44.7.) 83:4 125.1 | 166.8 | 208.5 | 250.2 | 291.9 | 333.6 | 375.3 
416 | 41.6 | 83.2 124.8 | 166.4 | 208.0 | 249.6 | 294.2 | 382.8 | 374.4 
4145. | 41.5 | 83.0 124.5 | 166.0} 207.5 , 249.0, 290.5 | 3832.0 | 373.5 
414 | 41.4] 82.8 124.2| 165.6 | 207.0 | 248.4 | 289.8 331.2 | 372.6 
413° | 41.3) 82.6 123.9 | 165.2} 206.5 | 247.8 | 289.1 | 330.4 | 371.7 
412 | 41.2 | 82.4 123.6 | 164.8 | 206.0 | 247.2 | 288.4 | 3829.6 | 370.8 
Ailes "44 A | 2.2 123.3 | 164.4 | 205.5 | 246.6 | 287.7 | 328.8 | 369.9 
410 | 41.0} 82.0 123.0 | 164.0! 205.0 | 246.0 | 287.0 | 828.0 | 369.0 
409 | 40.9| 81.8 {| 122.7 | 163.6 | 204.5} 245.4 286.3 | 327.2 | 368.1 
408 | 408] 81.6 122.4 | 163.2} 204.0 | 244.8 | 285.6 | 326.4 | 867.2 
407. | 40.7 | 81.4 122.1 | 162.8! 208.5 |} 244.2 | 284.9 | 325.6 | 366.3 
406°} 40.6 | 81.2 121.8 | 162.4 | 203.0) 24836) 284.2 | 824.8 | 365.4 
405 | 40.5} 81.0 121.5 | 162.0 | 202.5 | 243.0! 288.5 | 3824.0 | 364.5 
404 | 40.4 | 80.8 421.2 | 161.6 | 202.0 | 242.4 | 282.8 | 323.2 | 363.6 
403 |.40.3 | 80.6 420.9 | 161.2 | 201.5 | 247.8] 282.1 | 822.4 | 3862.7 
402 } 40.2! 80.4 120.6 | 160.8 | 201.0] 2412] 281.41 3821.6 | 361. 
401 | 40.1 | 80.2 120.3 | 160.4 | 200.5 | 240.6 | 280.7 | 820.8 | 360.9 | 
400 | 40.0 | 80-0 120.0 | 160.0} 200.0} 240.0: 280.0 | 820.0 | 360.9 | 
399 |.39.9 | 79.8 119.7} 159.6 | 199.5} 230.4 | 279.3 | 319.2 | 359.1 
398 | 39.8 | "9.6 | 119.4| 159.2 | 199.6) 288.8 | 278.6} 318.4 | 358.2 | 
397 | 39.7 | 79.4 | 119.1} 158.8 | 198.5.) 238.2) 277.9 | 317.6 | 357.3 
396 | 39.6 | 79.2) | 118.8! 158.4 | 198.0 | 287.6 | 277.2 316.8 | 356.4 
395 | 39.5 | 79.0 118.5 | 158.0 5 | 237.0 | 276.5 ' 316 0! 355.5 


| 


TABLE XXTV.—LOGARITHMS OF NUMBERS, 


RR STR aa ce cr 


| No. 110 L, 041.] [No. 119 L. 078, 


9 | Diff. 


a] 
ie 2) 


6 


or 


| N. | 0 pee 2 3 | 4 
| | Recaecacar | 

| | 

| 110 | 041393 | 1787 | 2182 | 2576 |, 2969 |] aga | 
| 1) 5323 | 5714 | 6105 | 6495 | 6885 || 7275 | weed | 8053 2 | 820! 590 
| 2%) 9218 | 9606 | 9993 i| - 

| 0380 | 0766 || 1153 | 1538 | 1924 | 2209 | c@o4 | 886 
3 | 053078 | 3463 | 3846 | 4230 | 4613 || 4996 5378 | 5760 | 6142 | 6524 | $83 
4 6905 | 7286 | 7666 | 8046 | 8426 || 8805 9185 | 9563 | 9942 


3755 | 4148 | 4540 | a9se | agg 
2 ») 


| 0820 | 3879 
4083 | 376 


5 | 060698 | 1075 | 1452 | 1829 | 2906 || e582 | eos | £333 | 3709 | 
: 7815 | 373 


| 
6 | 4458 | 4832 | 5206 | 5580 | 5953 || 6326 | 6699 OTL | 7448 | 
© | 8186 | 8557 | 8928 | 9298 | 9668 . — 
| ass 0088 | 0407 | 0776 | 1145 | 1514] 370 
\\ S*) 071882 | 2250 | 2617 | 2985 | 3352 || 3718 | 40gs 4451 | 4816 | 5182 | 366 
ee 9) 5547 | 5912 | 6276 | 6640 | 7004 |] 7368 | 731 | €094 | | 


ee 
Si 
or 
ee 


8819 | 363 
| 


PROPORTIONAL Parts. 


Dah =e ae <= 


| 
| Diff. 1 2 3 4 5 6 vi 8 9 
\| joe i ‘ ee ae 
395 | 39.5] 79.0 118.5 | 158.0 | 197.5 | 237.0 | 9765 316.0 | 355.5 
| 304-| 39.4] 78:8 118.2 | 157.6 | 197.0] 236.4 | 9275/8 | 815.2 | 354.6 | 
393. | 39.3 | 78.6 7.9) 157.2 | 166.5 | 235.8] on5 4 | 814.4 | 853.7 | 
892 | 39.2 | 78.4 117.6 | 156.8) 196.0} 235.9] 2974/4 | 313.6 | 352.8 
391 | 89.1 | 78:9 17.3 | 156.4 | 195.5 | 234.6] 93°77 | 312.8 | 251.9 
390 | 39.0! 8.0 V7.0 | 156.0; 165.0 | 984.0] 2973/0 | 812.0 | 351.0 
389 | 88.9] 77.8 116.7 | 155.6 | 194.5} 9&3 4 | ere | 811.2 | 850.4 
388. | 88.8 | 77.8 116.4 | 155.2! 194.0] F298] 9717/6 | 210.4 | 349.2 | 
887 | 88.7% | 274 116.1 | 154s 103.5 | 222.2 | 270.91 309.6 | 348.3 
386 | 88.6 | "7.9 115.8 | 154.4 | 198.0} 231.6 | 940.2 808.8 | 347.4 
885 "| 88.5 1° 77-0 115.5 | 154.0 | 152.5 | 231.0] 269.5 | 808.0 | 846.5 
384 | 38.41 76.8 115.2 | 158.6 | 162.0] 220.4] 268.8 | 207.2 | 345.6 
883 :| 38.3°| 76.6 114.9 | 158.2] 191.5 | 229.81 9684 306.4'| 344.7 
882 | 38.2-| 76.4 114.6 | 152.8 | 191.0} 209.9] 967.4 305.6 | 343.8 
381. | 88.1} 76.2 114.3 | 152.4] 190.5] 998.6 266.7 | 804.8 | 342.9 
880 | 88.0 | 46.0 114.0 | 152.0} 190.0] 9228 0 266.0 | 804.0 | 342.0 
379 | 87.9 | 75.8 113.7 | 151.6 | 189.5 | 2297.4] 965.3 | 808.2 | 841.1 
318 937,821) G56 113.4 | 151.2 | 189.0 | 226.8] 964'6 | 802.4 | 340.2 
Othe! 3%. Zale 724 113.1 | 150.8 | 188.5 | 226.2 | 963.9 | 301.6 | 339.3 
316 | 87.6.1 75.2 112.8 | 150.4 | 188.0] 22.6] 963.9 | 800.8 | 388.4 
375. | 87:5 | 75.0 112.5 | 150.0 | 187.5 | 225.0} 9262/5 | $00.0 337.5 
3ov4_ | 87.4 | 74:8 112.2 | 149.6 | 187.0 | 224.4} 961.8 | 299.2 | 336.6 
i 8784) S784 (7456 111.9 | 149.2 | 186.5 | 223.8 261.1 | 298.4 | 335.7 
| 87201 88.24 P4c4 111.6 | 148.8] 186.0} 223.2 | 960.4 | 297.6 | 334.8 
|: 3871-1 87.11. 74.2 111.3 | 148.4 | 185.5] 229.6 259.7 | 296.8 | 233.9 
870 | 387.0 | 74:0 111.0 | 148.0 | 185.0 | 222.0 | 9599 266.0 | 883.0 
| 369 | 36.94 73.8 110.7 | 147.6 | 184.5 | 201.4] $583 295.2 | 882.1 
| 868 186.8} 73.6 110.4 | 147.2} 184.0} 220.81 957/6 204.4 | 331.2 
i | 867 | 36.71 73:4 110.1 | 146.8} 183.5 | 980.9 | 256.9 | 293.6 | 330.3 
I 366 | 86.6 | 73.9 109.8 | 146.4 | 183.0] 219.6 | 956.2 | 292.8 | 829.4 
| 565 | 36.51 73.0 109.5 | 146.0} 182.5} 219.0] 955.7 } 292.0 | 328.5 
1] 364 | 86.4] 72.8 109.2} 145.6) 182.0] 218.4] 954.8 291.2 | 327.6 
Hil 363. -| 36.3.) 72-6 108.9 | 145.2 | 181.5) 217.8] 25471 | ‘990.4 | 306.7 
362 | 36.2] 72.4 108.6 | 144.8} 181.0] 217.9 | 953 4 | 289.6 | 825.8 
361 | 36.1 | 472.2 108.3 | 144.4 | 180.5] 216.61] 2959.7 288.8 | 824.9 | 
360 | 36.0! 72.0 108.0 | 144.0} 180.0] 216.0 959.0 | 228.0 | 324.0 | 
359 | 35.9 | 71.8 107.7 | 148.6} 179.5] 215.4] 251.3 287.2 | 223.4 
358 | 85.8 | 71.6 107.4 | 143.2 | 179.0} 214.8) 950.6 | 286.4 | 322.2 
OO7.31° 85.7 Vid 107.1 | 142 8) 178.5] 9214.9 | 949°9 285.6 | 221.2 | 
556 | 35.6 | 71.2 | 106.8 | 4 


142.4 178.0 213.6 249.2 284.8 | &€0. 


TABLE XXIV. 


No. 120 L. 079.) 


LOGARITHMS OF NUMBERS. 


[No. 134 L. 130, 


| Diff. 


Nokyt@e tthete & [8 +) 4 eS iin 9 
| | | | 
| : 
9 97918 954! C ~~, ———|] I 
120 079181 | 9543 | 9904 | O565 | o¢a6 || ooST | 1347 | 1707 | 2067 | 2426 | 360 
1 082785 | 3144 | 3503 8861 4219 | 4576 | 4934 | 5291 | 5647 | 6004 | 35% 
3| 6360 | 6716 | 7071 | 7426 | 7781 | 8136 | 8490 | 8845 | 9198 | 9552) 35d 
3| 9905 | — |——— | | | 
— 0258 | 0611 | 0963 | 1315 | 1667 | 2018 | 2370 | 2721 | 8071 | 852 
4 | 093422 | 3772 | 4122 | 4471 4820 || 5169 | 5518 | 5866 | 6215 6562) 349 
5 | 6910 | 7257 | 7604 | 7951 | 8298 |) 8644 | 8990 | 9385 | 9681 \———_| 
\- - 0026 | 346 
6 | 100371 | 0715 | 1059 | 1403 | 1747 |, 2091 | 2434 | 2777 | B119 | 8462 | 34s 
| 3804 | 4146 | 4487 4828 | 5169 || 5510 | 5851 | 6191 | 653i | 6871 | S4l 
8! 7210 | 7549 | 7888 S227 | 8565 || 8903 | 9241 | 9579 | 9916 |— 
.| | | eee| 0253 | 3838 
9 | 110590 | 0926 | 1263 | 1599 | 1934 |) 2270 | 2605 | 2940 | 8275 | 8609 | 335 
130 3943 | 4277 | 4611 | 4944 | 5278 || 5611 | 5943 | 6276 | 6608 | 6940 | 338 
1 | 771 | 7603 | 7934 8265 | 8595 || 8926 | 9256 | 9586 | 9915 | | 
— | | | 0245 | 330 
| | 120574 | 0903 | 1231 | 1560 | 1888 || 2216 | 2544 | 2871 | 8198 | 8525 | 328 
| 31} 3852 | 4178 | 4504 | 4830 | 5156 || 5451 | 5806 6131 | 6456 | 6781 | 825 
4| 7105 | 7429 | 7758 | 8076 | 8399 || 8722 | 9045 | 9368 | 9690 
| 13 | | | ooi2 | 82 
| 
PROPORTIONAL PARTS. 
| 
| Diff.| 1 2 3 4 5 se eee 8 9 
55. | 35.5 | 71.0 | 106.5 | 142.0 | 177.5 | 213.0) 248.5) 284.0 | 319.5 
a4 | 35.4| 70.8 | 106.2} 141.6] 177.0 | 212.4 | 247.8) 283.2 | 318.6 
gna! | 35.3 | 70.6 | 105.9 | 141.2 | 176.5 | 211.8 | 247.1) 262.4 | 3iv.% 
359 | 35.21 70.4 | 105.6 | 140.8] 176.0 | 211.2 | 246.4 | 281.6 | 316.8 
ost | 35.1 70.2 | 105.3} 140.4 | 175.5 | 210.6) 245.7 | 280.8 | 315.9 
350 | 35.0! 70.0 | 105.0} 140.0] 175.0 | 210.0 | 245.0 | 280.0 | 315.0 
| 349 | 34.91 69.8 | 104.7 | 139.6] 174.5 | 209.4) 244.3) 279.2 | 314.1 
| 348 | 34.8] 69.6 | 104.4] 139.2] 174.0 | 208.8) 243.6) 278.4 | 818.2 
a4z | 34.7 | 69.4 | 104.1 | 188.8) 173.5) 208.2 | 242.9) 277.6 | 312.3 
346 | 34.6 | 69.2 | 103.8 | 188.4 | 173.0) 207.6 | 242.2 | 206.5 | 311.4 
| | | 
ain | 34.5 | 69.0 | 108.5 | 138.0] 172.5] 207.0) 241.5 | 276.0 | 310.5 
} 344 | 84.4 | 68.8 | 108.2} 137.6) 172.0 206.4 | 240.8 | 275.2 | 309.6 
343 | 34.3 | 68.6 | 102.9| 137.2 | 171.5 | 205.8) 240.1 | 274.4 | 308.7 
342 | 34.2 | 68.4 | 102.6| 136.8] 171.0} 205.2 | 239.4) 273.6 | 307.8 
B41 | 34.1 | 68.2 | 102.3 | 186.4 | 170.5 | 204.6 | 238.7 | 212.8 | 806.9 
310 | 34.0 | 68.0 | 102.0| 136.0 | 170.0 | 204.0 | 238.0 | 272.0 | 306.0 
939 | 33.9 | 67.8 | 101.7 | 135.6 | 169.5 | 203.4 | 287.3 | 271.2 | 306.2 
398 | 33.8 | 67.6 | 101.4| 135.2| 169.0] 202.8} 236.6) 270.4 | 304.2 
337 | 33.7) 67.4 | 101.1} 134.8 | 168.5 | 202.2) 235.9) 269.6 | 308.3 
335. | 33.6 | 67.2 | 100.8} 134.4] 168.0 | 201.6 | 235.2) 268.8 | 302.4 
| 385 | 98.5 | 67.0 | 100:5:| 134.0) 167.5.) 201.0 | 230.5 268.0 | 301.5 
| 334 | 83.4 | 66.8 | 100.2) 133.6 167.0 | 200.41 233.8) 267.2 | 300.6 
| 333 | 33.3 | 66.6 99.9 | 133.2 | 166.5 | 199.8 | 2838.1 | 266.4 | 299.7 
339 | 33.2! 66.4 | 99.6| 182.8 | 166.0| 199.2 | 232.4 | 265.6 | 298.8 
331 | 33.1 | 66.2 | 99.3| 182.4] 165.5 | 198.6 | 281.7 | 264.8 | 297.9 
339 | 33.0! 66.0 | 99.0} 132.0! 165.0 | 198.0) 231.0 | 264.0 | 297.0 
329 | 32.9 | 65.8 98.7 | 131.6 | 164.5) 197.4 | 230.3 | 263.2 | 296.1 
308 | 32.8| 65.6 | 98.4 | 1381.2 | 164.0 | 196.8 | 220.6 | 262.4 | 295.2 
327 2.7 | 65.4 98.1 130.8 163.5 | 196.2} 228.9 | 261.6 | 294.3 
326 | 32.6 | 65.2 97.8 | 130.4 | 163.0 | 195.6 | 228.2} 260.8 | 293.4 
2.5 | 65.0 97.5 | 130.0 | 162.5 | 195.0 | 227.5 | 260.0 | 292.5 
241 64.8 | 97.2 | 129.6] 162.0] 194.4 | 226.8 | 259.2 | 201.6 
2.3 | 64.6 96.9 | 129.2} 161.5, 193.8 926.1 | 258.4 | 290.7 
2.2 | 64.4 96.6 | 128.8! 161.0! 193.2 | 225.4] 257.6 | 289.8 


No. 135 L. 130.] 


TABLE XXTV.—LOGARITHMS OF NUMBERS. 


[No. 149 L. 1%, 


1 


2 


| 0655 
3858 


| 7037. | 


| 0 
41% 
7 


f 
‘ | 
B54. 


97 


1298 | 1619 || 1939 | 2260 
| 4496 | 4814 1} 5133 | 5451 
7671 | 7987 || 8303 | 8618 | 


2900 
6086 


9249 | 


| 0508. | 
| 8639 | 
| 6748 | 
| 9885 


0822 | 11 
8951 | 42 
7058 | 73 


36 || 1450 | 1763 
63 || 4574 | 4885 
367 || 7676 | 


| 2389 
| 5507 


| 8603 


2594 
| 5640 
| 8664 


2900 | 


5943 
8965 


9266 | 9567 || 9868 | 


| 0142 | 0449 || 0756 | 1063 
3205 | 3510 | 
6246 | 6549 || 6862 | 7154 


8815 


| 1676 
| 4728 


C159 | 


| 1667 


| 4650 
7613 


1967 
4947 
7908 


| 2266 | 2564 
5244 | 5541 || 5888 | 6134 
8203 | 8497 || 8792 


2863 | 3161 


9086 


| 0769 | 
| 38758 | 
| 6726 


| 9674 


0555 
3478 


0848 
3769 


| 1141 | 1434 || 1726 
4060 | 4351 || 4641 | 4932 


2019 


| 


26083 


5512 


PROPORTIONAL PARTS, 


i a) 


1D 


Cos 


t 


=) 


et ec CoH 


DDS 


Sr 2 


He O 


- 


ww Oo 


oor 


COM HE -3 O to 


oO 


L950 Hwa 


ee 


CO he = 


w on 


© 


Oo 


WHIP IOW HDowordueshwoO es 


2X 

© & 
= & 
wos 


o] 
2] 
~ 


pare 
Noi) 


~J iH ooo 


S 
Su -2 
Cow © 


rw) 
Nort Le Cora 


rar) 
He rt GO OTAICO OC 


2 © 
Parco 
eo CO OO 

o- 


i] 
DAs 


4 5 6 
128.4 | 160.5 |] 192 
128.0 160.0 192. 
127.6 159.5 191. 
127.2 159.0 | 190 
126.8 158 5a 19 
126.4 58.0} 189 
126.0 | 157.5 | 189 
125.6 | 157.0 | 188 
125.2 156.5.) 18% 
124.8 156.0 187 
124.4 | 155.5 186 
124.0°| 155.0}. 186 
123.6 | 154.5.| 185 
122.8 | 158.5 84. 
122.40) T58.0ab 183° 
122.0 | 152.5} 183. 
121.6 | 152.0 } 182. 
12132 15155 181 
120.8 152.-Oe}) 187 
120.4} 150.51 180 
120.0 150.0 80 
119.6,| 149.5.) 179. 
119.2 1490+ 178. 
118.8; 148.5 i 78. 
118.4 148.0 17%. 
118.0 147.5 Lik. 
117.6 TAG Ont R622 
117.2 | 146.5 175: 
116.8 146.0 175. 
116.4 145.5 174. 
T1650 145.05) 77a 
Aid Gay” 144.5 13. 
115.2 | 144.0} 179, 
114.8} 1438.5 | 179. 
114.4.) 143.0) (174. 


WOOWO 


© OIA Emod 


261. 
260. 
259. 
258. 
a 


ony 


| && 


DBI wuUP OW 


Hoo 


S 
Nw 


=) 


i OD dD 


TABLE XXIV.—LOGARITHMS OF NUMBERS. 


ae 


| No. 150 L. 176.] [No. 169 L. 230, 


Oo] wl a] a | 4] s 6) 7 | s.| 9 | pis 


176091 | 6381 ; 6.70 | 6959 | 7248 || 75386 | 7825 | 8113 | 8401 | 8689 | 289 
| 8977 | 9264 | 9552 | 9839 = | ee Pe 

: 0126 || 0413 | 0699 | 0986 | 1272 | 1558 | 287 
181844 | 2129 | 2415 | 2700 | 2985 || 8270 | 8555 | 38389 | 4123 | 4407 | 285 
| 4691 | 4975 | 5259 | 5542 | 5825 || 6108 | 6591 | 6674 | 6956 | 7239 | 283 
7521 | 78uB | 8084 | 8366 + $647 || 8928 | 9209 | 9490 | 9771 


| 4 | 0051 | 281 
| 190332°| 0612 | 0892 | 1171 | 1451 || 1780 | 2010 | 2289 | 2567 | 2846 | 279 
| 8125 | 3403 | B681 | 3959 | 4237 || 4514 | 4792 | 5069 | 5346 | 5623 | 278 i 
| 5900 | 6176 | 6458 | 6729 | 7005 || 7281 | 7556 | 7882 | 8107 | 8882 | 276 i 
8657 | 8932 | 9206 | 9481 | 9755 || | HH 
| || 0029 | 0303 | 0577 | 0850 | 1124 | 274 | 
201397 | 1670 | 1943 | 2216 | 2488 || 2761 | 8033 | 8805 | 3577 | 3848 | 272 
4120 | 4301 | 4663 | 4934 | 5204 || 5475 | 5746 | 6016 | 6286 | 6556 | 271 
6826 | 7096 | 7365 | 7634 | 7904 || 8173 | 8441 | 8710 | 8979 | 9247 | 269 
9515 | 9783 : | ‘ i 
| 0051 | 0319 | 0586 || 0853 | 1121 | 1888 |. 1654 | 1921 | 267 | 
212188 | 2454 | 2720 ; 2986 | 3252 || 2518 | 3783 | 4049 | 4314 | 4579 | 266 | 
4844 | 5109 | 5873 | 5638 | 5902 || 6166 | 6420 | 
7484 | 7747 | 8010 | 8273 | 3798 | 9060 
| | - — 
220108 | 0370 | 0631 | 0892 | 1153 |) 1414 | 1675 | 1936 | 2196 | 2456 | 261 i 
2716 | 2976 | 3236 | 3496 | 8755 |) 4015 | 4274 | 4533 | 4792 | 5051 | 259 
5309 | 5568 | 5826 | 6084 | 6342 || 6600 | 6858 | 7115 | 7372 | 7630] 258 


6694 | 6957 | 7221 264 i 
9323 | 9585 |. 9846 | 262 Hl 


® 
Or 
Co 
C3 
ee 
~> 
OO 


om es 
' We, Soe = ict a} 2 
COOID TRO WHOS © DIM Bww HS] = 


| I 
7887 | 8144 | 8400 | 8657 | 8913 |) 9170 | 9426 | 9682 | 9938 | —— I 
| Seay | | | | 0193 | 256 | 
PROPORTIONAL PARTS, | 
Bestia: 4] x ie zB Aim 
BE Aad 1 testes Cadet ok ee (atehs PR aie ie! 
| | A r | Set UE | | i 
| | 285 | 28.5] 57.0.| 85.5 | 114.0] 142.5] 171.0] 199.5 | 228.0 | 256.5 
| | 284- | 28.4 | 86.8 | 85.2 | 113.6 | 142.0} 170.4} 198.8 | 227.2 | 255.6 
| 283 | 28.3} 56.6 | 84.9 | 118.2] 141.5] 169.8 | 198.1 | 226.4 | 254.7 
| 282 | 28.2 | 56.4 | 84.6 | 112.6| 141.01 169.2 | 197.4 | 225.6 | 258.8 ' 
| 281 | 28.1 | 56.2 | 84.3 | 1124] 140.5 | 168.6 | 196.7} 224.8 | 252.9 | 
| 280 | 28.0} 56.0 | 84.0 | 112.0] 140.0| 168.0] 196.0 | 224.0 | 252.0 et 
279 | 27.9) 55.8 | 83.7 | 111.6 | 139.5 | 167.4 | 195.3 | 223.2 | 251.1 
| 278 | 27.8 | 55.6 | 88.4 | 111.2] 139.0} 166.8) 194.6 | 222.4 | 250.2 } 
a7 | 27.7 | 55.4 | 88.1 | 110.8 | 138.5} 166.2] 193.9 | 221.6 | 249.3 , 
: 276 | 27.6) 55.2 | 82.8 | 110.4] 138.0°| 165.6 | 193.2 | 220.8 | 248.4 I 
: | 975 | 97.5 | 55.0 | 82.5 | 110.0| 187.5} 165.0! 192.5 | 220.0 | 247.5 } 
"ora | 27.41 54.8 | 82.2 | 109.6 | 137.0} 164.4} 191.8 | 219.2 | 246.6 } 
973 | 27.3) 54.6 |) 81.5 109.2 | 136.5 | 163.8! 191.1 | 218.4 | 245.7 | i 
| 272 | 27.2; 54.4 | 81.6 | 108.8] 186.0} 163.2} 190.4; 217.6 | 244.8 ; 
| | 271 | 27.1) 54.2 | 81.8 | 108.4 | 185.5) 162.6! 180.7 | 216.8 | 248.9 
| | 270 | 27.0] 54.0 | 81.0 | 108.0! 185.0] 162.0) 189.0) 216.0 | 2438.0 
| 1 269 | 26.9! 53.8 | 80.7 | 107.6) 134.5 | 161.4] 188.3! 215.2 | 242.1 
| 268 | 26.8! 53.6 | 80.4 | 107.2| 184.0! 160.8) 187.6 214.4 | 241.2 
267 | 26.7) 53.4 | 80.1 | 106.8] 188.51 160.2} 186.9} 218.6 | 240.% 
| 266 | 26.6 | 53.2 | 79.8 | 106.4 | 133.0 | 159.6 | 186.2) 212.8 | 220.4 | 
| 265 | 26.5 | 53.0 | 79.5 | 106.0| 132.5) 159.0} 185.5 ' 212.0 | 238.5 | | 
: 264 | 26.4| 52.8 | 79.2 | 105.6] 182.0) 158.4 | 184.8) 211.2 | 237.6 | 
) 263' | 26.3 | 52.6 | 78.9 | 105.2) 181.5 | 157.8) 184.1 210.4 | 236.7 
| 262 | 26.21 52.4 | 78.6 | 104.8] 181.0] 157.2 | 183.4 | -209.6 | 285.8 
| 261 | 26.1| 52.2 | 78.3 | 104.4 | 130.5) 156.6 | 182.7 | 208.8) 2384.9 
| 260 | 26.0; 52.0 | 78.0 | 104.0 | 120.0 156.0 | 182.0 | 208.0 | 234.0 
| | 259 | 25.9; 51.8 | 77.7 103.6 | 129.5 155.4} 181.38 | 207.2 | 288.1 
, 258 | 25.8) 51.6 | 77.4 | 108.2 | 129.0) 154.8 | 180.6 | 206.4 | 282.2 
DT | 95.9% S40 F714 102.8 | 128.5) 154.2 | 179.9 | 205.6 | 231.8 
P56 @)25.671 251.2 76.8 |-102.4 | 128.0; 153.6) 179.2 | 204.8) 230.4 
255 125.51 51.0 | 76.5 | 102.0) 127.5) 153.0} 178.5 | 204.0 | 229.5 | 


No. 170 L. 230.] 


TABLE XXIV.—LOGARITHMS OF 


NUMBERS. 


[No. 189 L. 278. 


0 


1 


=i 
~) 


2 


3. | 64 


5 


6 


8 


| 
| 
| 
} 


9 


Diff. 


‘0 | 230449 | 0704 | 0960 | 1215 | 1470 || 1724 | 1979 | 2284 | 2488 | 2742 | 255 
1} 2996 | 3250 | 8504 | 8757 | 4011 || 4264 | 4517 | 4770 | 5023 | 5276 | 253 
2] 5528 | 5781 | 6083 | 6285 | 6537 || 6789 | 7041 | 7292 | 7544 | 7795.| 252 
3 | 8046 | 8297 | 8548 | 8799 | 049 || 9299 | 9550 | 9800 |———| 
| | | 0050 | 0300 | 250 
4 | 240549 | 0799 | 1048 | 1297 | 1546 || 1795 | 2044 | 2293 | 2541 | 2790 | 249 
5 | 3038 | 8286 | 3534 | 8782 | 4030 || 4277 | 4525 | 4772 | 5019 | 5266 | 248 
6 | 5513 | 5759 | 6006 | 6252 | 6499 || 6745 | 6991 37 | 7482 | 7728 | 246 
7 | 7973 | 8219 | 8464 | 8709 | 8054- || 9198 | 94438 7 | 99382 ar 
2 —| 0176-! 245 
8 | 250420 | 0664 | 0908 | 1151 | 1395 || 1638 | 1881 | 2125 | 2368 | 2610 | 243 
9 | 2853 | 3096 | 3338 | 3580 | 3822 || 4064 | 4306 | 4548 | 4790 | 5031 | 242 
180 | 5273 | 5514 | 5755 | 5996 | 6237 || 6477 | 6718 | 6958 | 7198 | 7439 | 241 
1 7679 | 7918 | 8158 | 8398 | 8637 || 8877 | 9116 | 9355 | 9594 | 98383 | 239 
2 | 260071 | 0310 | 0548 | 0787 | 1025 || 1263 | 1501 | 1739 | 1976 | 2214 | 288 
3 | 2451 | 2688 | 2925 | 3162 | 3399 | 3636 | 3873 | 4109 | 4346 | 4582 | 237 
4| 4818 | 5054 | 5290 | 5525 | 5761 || 5996 | 6232 | 6467 | 6702 | 6937 | 235 
5 | 7172 | 7406 | 7641 | 7875 | 8110 || 9344 | 8578 | 8812 | 9046 | 9279 | 234 
6 9513 | 9746 | 9980 | | 
0213 | 0446 || 0679 | 0912 | 1144 | 1377 | 1609 | 233 
7 | 271842 | 2074 | 2306 | 2538 | 2770 || 8001 | 3233 | 3464 | 3696 | 3927 | 232 
8 | 4158 | 4389 | 4620 | 4850 | 5081 || 5311 | 5542 | 5772 | 6002 | 6232 | 280 
9 | 6462 | 6692 | 6921 | 7151 | 7380 '| 7609 | 7828 | 8067 | 8296 | 8525-| 229 
PROPORTIONAL PARTS. 
Diff.| 1 2 3 4 5 6 7 8 9 
255 | 25.5] 51.0 | 76.5 | 102.0] 127.5] 158.0 | 178.5 | 204.0 | 229.5 
254 | 25.4.) 50:8} 76.2 9) 101.6) 127.0 | 152.4:) 17728 |> 208.9 228.6 
QS) 25. Bc 5O.6ech Hah TORQ: -126.5)|) 161 7Bsle 177 dd Ge 202. aah, Saraem. 
252 | 25.2) 50.4 | 75.6 100.8 | 126.0 | 151.2} 176.4 | 201.6 | 226.8 
251 | 25.15). 50.2 4 58>) 100.4 + 125.5:| 150.6.) 175.7 |, 200.8 | 225.9 
250 | 250| 50.0 | %5.0 100.0 | 125.0} 150.0 | 175.0 | 200.0 | 225.0 
219 | 24.9) 49.8 | 74.7 99.6} 124.5 | 149.4] 174.3 | 199.2 | 224.1 
248 | 24.8| 49.6 | 74.4 99.2 | 124.0] 148.8} 173.6 | 198.4 | 223.2 
Bay Ay) AGA | aed 98.8 | 128.5 | 148.2 | 172.9 | 197.6 | 222.8 
246 | 24.6 | 49.2 | 73.8 98.4 | 123.0] 147.6] 172.2 | 196.8 | 221.4 
245 | 24.5 | 49.0 | 8:5 98.0 | 122.5 | 147.0} 171.5 | 196.0 | 220.5 
244 | 24.4] 48.8 | 73.2 97.6 | 122.0] 146.4} 170.8 | 195.2 | 219.6 
243 | 24.8 | 48.6 72.9 97.2) 121.5) 145.8}. 170.1.| 194.4 | 218.% 
242 | 24.2] 48.4 | 72.6 96.8 | 121.0] 145.2] 169.4 | 193.6 | 217.8 
BAdhe) 24.49): 489-2) 7253 96.4 | 120.5 | 144.6] 168.7 | 192.8 | 216.9 
240 | 24.0 | 48.0 | 72.0 96.0 | 120.0} 144.0] 168.0] 192.0 | 216.0 
239 | 23.9 | 47.89] 71:7 95.6). 119.5) 143.4 | 167.8 | 191.2 | 215.1 
938 | 23.8! 47.6 | 71.4 95.2 | 119.0], 142.8] 166.6 | 190.4 | 214.2 
i EG ev ae ev Se Sa ae | 94.8 | 118.5 | 142.2] 165.9 | 189.6 | 213.3 
236 | 23.6] 47.2 70.8 94.4] 118.0] 141.6 | 165.2 | 188.8 | 212.4 
tee ea: 5 Tt 47.0" f 67085 94.0 | 117.5 | 141.0) 164.5 | 188.0 | 211.5 
234 | 23.4] 46.8 | 70.2 93.6 | 117.0 | 140.4] 163.8 | 187.2 | 210.6 
9 3 ale eS Was 46.6 69.9 93.2 116.5: . 18928 163.1 | 186.4 | 209.17 
232 | 23.2} 46.4 | 69.6 92.8 | 116.0 | 139.2 | 162.4 | 185.6 | 208.8 
231 | 23.1 | 46.2 | 69.3 92.4 | 115.5 | 138.6 | 161.7 | 184.8 | 207.9 
230 | 23.0] 46.0 | 69.0 92.0 | 115.0] 138.0] 161.0] 184.0 | 207.0 
229 | 22.9] 45.8 | 68.7 91.6] 114.5 | 187.4} 160.3 |. 183.2 | 206.1 
228 | 22.8| 45.6 | 68.4 91.2 | 114.0] 186.8] 159.6 | 182.4 | 205.2 
227 22.7 45.4 68.1 90.8) 118.5 136.2 158.9 181.6 | 204.3 
226 | 22.6 | 45.2 | 67.8 90.4 | 113.0] 185.6 | 1582} 180.8 | 203.4 


TABLE XXTV.—LOGARITHMS CF NUMBERS. 


| No. 190 L. 278.] 


Sa a 
| 


owe | 
199 | 278754 | 8982 


ase 
U 
Y 
> 


| 1261 | 1488 | 


[No, 214 L, 232, | 
+ hee | 


9 


2 3201 | 3527 
3 5557 | 5782 
4 | [7802 | 8026 
5 | 290085 | 0257 | 
6 2256 | 2478 | 
v 4466 | 4687 | 


8 | 6665 | 6884_| 
8853 | 9071 | 


9507 


200 | 301030 | 1247 
1 | 3196 | 38412 


| 2} 5351 | 5566 | 
| 3 | 7496 | 7710 
4} 9630 | 9843 | 


| 
| 5 | 811754 | 1966 | 
|} 3867 | 4078 
| 5970 | 6180 | 
| | 8 | 8063 | 8272 | 


ID 


| 1681 
| a04d4 


5996 


$1387 


83851 


| O268 
| 2589 | 
| 4499 


6599 
8689 


| 0481 


2600 
4710 
6809 


Z 
8898 


5760 | 
W854 


Q2: 
9958 


| 320146 


=) 
co 
Ol 
a 


2219 | 2426 
| 1| 4282 | 4488 
| | 2 | 6386 | 6541 | 


3 |  8880-| 8583 | § 


| O769 | 


IQS 
WO 


4899 | 


6950 
8991 


0977 | 
8046 


5105 


1155 | 


9194 


4 | 330414 | C617 | 0819 | 


1022 


1225 


PROPORTIONAL PARTS. 


2012 | 


4nhnw 


407% 


6131 


| 8176 


0211 | 
2230 | 


en. | b>} 42 


oo 


Rey 


| z | was 

| 205 | 22.5 | 45.0 

| 924 | 224] 44.8 

| 993 |92.3| 44.6 

| 999 | 999) 44.4 
294 | 2.1) 44.2 
220 | 22.0] 44.0 
219 | 21.9| 43.8 


rw 
a 
re) 

t COC 
IS 
Ww 
(or) 


wO 0 
— Re 
C22 
2 2 
eee 
ee 
aN 
Ww YW 
2S He 


215 1 2e 
ube = Ot et 42.3 
| | 218 21.3 | 2.6 
212 21.2 | 2.4 
211 el MA Ale 
| 210 ALO 42 .( 
| 209 20. 41.8 
i} 208 20.8 41.6 
207 20.7 41.4 
206 20.6 41.2 
205 20.5 4d .0 
£04 20.4 40.8 
2038 20.3 40.6 
202 | 20.2 40.4 


Cwo°iwore W-2Oco 


5 eS 
OWS 


owce 
Oooce 


t 


oo 
CHOW OW ROT 


OROOWBORO 


Wwe wwwww 


It ..St 


CU 


at 


oo 


WwWwww 


~j =} 


Wo LO OO 


Oo 


ee ee on oe on eed 


S Cr Coa 


SOAS % 


It 


oo 


SO WWWWWW 


} PARED CD COWS HE Cr 


Bek fk fk ed Ped Pe Pe 


| 


} 


= 


(or) 


3 ris 
ON PR OUWD AWORKH Or 


| Pe COOTWw< 


— — — 
Rok Oro <3 
9S OG KW 


= 
cw) 
ee 2 


po 


vO 


co 2 


2 Cd OT he 


CO 2 et 


Or 


SONG 


2 


Go 


He CIM HO 


OO wo oO 


0 


bo 


3044 | 32 


TABLE XXTV.—LOGARITHMS OF NUMBERS. 


pas cS See Fe ae 
No, 215 L, 832.] 


| 8417 


8649 


[No. 239 L. 880. 


7 | 8 


| 4051 


2028 | 2225 


332438 3850 
4454 | 50ET | || 5458 | 5658 | 5859 | 6059 
6460 7060 7459 | 7659 | 7858 | 8058 
8456 | 9054 9451 | 9650 | 9849 |——— 
| | 5: 41 | 0047 

340444 *| 1039 | '| 1435 | 1682 | 1830 
2423 3014 3409 | 3606 | 8802 | 8999 | - 
4392 4931 | 5374 | 5570 | 5766 | 5962 
6353. | 6939 7330 | 7525 | 7720 | 7915 
8305. | | 8889 9278 | 9472 | 9666 | 9860 

350248 0829 | 1216 | 1410 | 1603 | 1796 | 
£188 | 2761 | 8147 | 3839 | 3532 | 3724 
4108 4685 5068 | 5260 | 5452 | 5643 
6026 | | 6599 | 6981 | 7172 | 7363 | 755 
7935 8506 8886 | 9076 | 9266 | 9456 
9835 | / 

—— 0404 || 0783 | 0972 | 1161 | 1850 

361728 | 2294 | || 2671 | 2859 | 3048 | 3236 


3612 | 88 
5488 | 56 
7356 | % 
9216 


4176 


+ 6049 | 


“915 


9772 


lin | 


4551 


8287 


| 4739 
6423 


6610 
8473 


4926 
6796 
8659 


5113 
6983 
8845 


371068 
2912 
4748 
6577 


8398 


38 


| 1622 


3464 


5298 


7124 
8943 


0148 
1991 
8831 


5664 | 
7488 | 
| 9306 


0328 


2175 


4015 
5846 
7670 


9487 


PROPORTIONAL PARTS. 


0513 
2360 
4198 
6029 
7852 


9668 


0698 


2544 | 272 


4382 
6212 | 
8034 
9849 


J dD 
SS 
at) 


NDBWOOHKWN BROONAIDOOSO! 
SDOSSS | a 


MAM BAD 
WDOHPRNOWO 


Nard raIIW 
OH WIOWSSD 


jor) 


Oo OT 


Cs 


iO C2 ODS 


co 


DoHnww mE 


D-F MWOBWDWO?¢ 
— 
SO 66'S: 
oo 


for) 


et ee ee ee ee Br oe 


WNL KK oror 


STE Wd 3 3 SS 


rt 0o 


101. 
100. 


CO eS 
Wolo) 


re) © 
| . - <a . 
ASNOMOSEH CMOMOMON SDAOMOMOMS 


pe 


| 
| Co 
| 


fet 
We 
=) 


WO ROAWNDhoOaw 


loon) 


ro 


AWORO Mr 


pod 
Ho 
are 
— 


—_ 

sy) 

t CO 
DOWOAWO TH 


=e ge 
oo 9 
ev) Hm 

wo tke 


3 


Set LAW OD” HUT DAF WSO 


~ 


=~ >=) => 


th oF 


TABLE XXIV.—LOGARITHMS OF NUMBERS, 


No, 240 L. 380.] [No. 269 L, 431. 


Mein oe | boli a {owl ae lbs 6 | a] 8 | 9 [pie 


240 | 380211 | 0392 | 0573 | 0754 | 0934 |) 1115 | 1296 | 1476 | 1656 | 1887] 181 


Ae) 2017 | 2197 | 2377 | 2557 | 2787 || 2917 | 3097 | 38277 | 8456 | 3636 | 180 


; 3815 | 3995 | 4174 | 43853 | 4533 || 4712 | 4891 | 5070 | 5249 | 5428 | 179 

3) 5606 | 5785 | 5964.) 6142 | 6321 || 6499 | 6677 | 6856 | 7034 | 7212 178 

| 4 7390 | 7568 | 7746 | 7924 | 8101 || 8279 | 8456 | 8634 | 8811 | 8989 178 
| 5 9166 | 9343 | 9520 | 9698 | 9875 || 


6 | 390935 | 1112 | 1288 
7 | 2697 | 2873 | 3048 | 3224 | 3400 || 3575 | 3751 | 3926 | 4101 | 4277 | 176 

8 | 4452 | 4627 | 4802 | 4977 | 5152 || 5326 | 5501 | 5676 | 5850 | 6025 | 175 Ht! 
9 | 6199 | 6374 | 6548 | 6722 | 6896 || 7071 | 7245 | 7419 | 7592 | 7766 | 174 


250 | 7940 | 8114 | 8287 | 8461 | 8634 |! 8808 | 8981 | 9154 | 9328 | 9501 | 17% 
| 9674 | 9847 | = 


i} 


0051 | 0228 | 0405 | 0582 | 0759 | 177 i 
1464 | 1641 || 1817 | 1993 | 2169 | 2345 | 2521 | 176 


_—_—. 0020 | 0192 | 0365 || 0538 | 0711 | 0883 | 1056 | 1228 173 
2 | 401401 | 1578 | 1745 | 1917 | 2089 || 2261 | 2438 | 2605 | 28777 | 2949 172 | 
3 | 8121 | 8292 | 3464 | 3635 |.3807 || 38978 | 4149 | 4320 | 4492 | 4663 171 f| 
£ 4834 | 5005 | 5176 | 5346 | 5517 |) 5688 | 5858 | 6029 | 6199 | 6370 171 i} 
5 6540 | 6710.| 6881 | 7051 | 7221 || 7291 | 7561 | 7731 | 7901 | 8070 170 il 
6 | 8240 | 8410 | 8579 | 8749 | 8918 || 9087 | 9257 | 9426 | 9595 | 9764 169 i} 
th 9933 ii 


| 0102 | os71 | 0440 | 0609 |) 0777 | 0946 | 1144 | 1288 | 1451 i 
8 | 411620 | 1788 | 1956 | 2124 | 2293 |) 2461 | 2629 | 2796 | 2964 | 3132 | 168 \ 


art 
o> 
vo) 


9 3300 | 3467 | 3685 | 8803 | 3970 || 4187 | 4305 | 4472 | 4639 | 4806 167 
260 4973 | 5140°| 5307 | 5474 | 5641-1; 5808 | 5974 | 6141 | 6308 | 6474 | 167 
1 6641 | 6807 | 6973 | 7189 | 7306.) 7472 | 76388 | 7804 | 7970 | 8135 166 i 
2 8301 | 8467 | 8633 | 8793 | 8964 |) 9129 | 9295 | 9460 | 9625 | 9791 165 i 
3 9956 | ee | 
\ 

| 


| 0121 | 0286 | 0451 | 0616 781 | 0945 | 1110 | 1275 | 1489 165 
4 | 421604 | 1768 | 1933 | 2097 | 2261 || 2426 | 2590 | 2754 | 2918 | 3082 164 
5 3246 | 3410 | 3574 | 3737 | 3901 || 4065 | 4228 | 4392 | 4555 | 4718 164 
5371 | 5534 


6 4882 | 5045 | 5208 


5697 | 5860 | 6028 | 6186 | 6349 163 | 
f 


7 | 6511 | 6674 | 6836 | 6999 | 7161 || 7324 | 7486 | 7648 | 7811 | 7973 | 162 
8 | 8135 | 8297 | 8459 | 8621 | 8783 |) 8944 | 9106 | 9268 | 9429 | 9591 | 162 


9| 9752 | 9914 i | 
43 | 0075 | 0236 | 0398 | 0559 | 0720 | 0881 | 1042 | 1203.| 161 
i 
PROPORTIONAL PARTS, I 
hi 
| 
: - le 
| pie.| 1 | 2 3 4 5 6 " ea a: I 
| | } 
178 | 17.8| 35.6 | 53.4 | 71.2 | 89.0 | 106.8) 124.6} 142.4 | 160.2 I 
177 | 17.7| 35.4 | 53.1 | 70.8 | 88.5 | 106.2] 123.9 | 141.6 | 159.8 | 
176 | 17.6] 35.2 | 52.8 | 70.4 | 88.0 | 105.6] 128.2] 140.8 | 158.4 } 
Pe 17 5 | 85.04") 52.5.8) 70.0 7.5 | 105.0) 122.5 | 140.0 | 157.5 | 
174 | 17.4| 34.8 | 52.2 | 69.6 | 87.0 | 104.4] 121.8 |. 139.2 | 156.6 
173 | 17.3] 34.6 | 51.9 | 69.2 | 86.5 | 108.8) 121.1 | 138.4 | 135.7 
172 | 17.2] 34.4 | 51.6 | 68.8 | 860 | 103.2] 120.4] 187.6'| 154.8 
| 171 | 17.1] 34.2 | 51.8 } 68.4-} 8.5 | 102.6] 119.7] 186.8 | 153.9 
: 170 | 17.0} 34.0 | 51.0 | 68.0) 85.0 | 102.0} 119.0] 186.0.| 153.0 
169 |,16.9°) 33.8 | 50.7 | 67.6-4 81.5 | 101.4] 118.3] 185.2} 15214 
163 | 16.8} 33.6 | 50.4 | 67.2 + 84.0 | 100.8} 117.6 | 134.4 | 151.2 
167 | 16.7] 383.4 | 50.1 | 66.8 + 83.5 | 100.2] 116.9 |. 183.6] 150.3 
166 | 16.6] 33.2 | 49.8 | 66.4 } 83.0 99.6 | 116.2 | 132.8 | 149.4 
165 | 16.5] 33.0 | 49.5 | 66.0 | 82.5 99.0] 115.5 | 132.0 | 148.5 
164 | 16.4] 32.8 | 49.2 | 65.6 4 82.0 98.4} 114.8] 131.2 | 147.6 
163 | 16.3} 32.6 | 48.9 | 65.2 |} 81.5 97.8| 114.1] 130.4 | 146.7 
162 | 16.2) 32.4 | 48.5 | 64.8-'F 81.0 97.2 | 118.4] 129.6 | 145.8 
161 | 16.1] 32.2 | 48.3 | 64.4 } 80.5 96.6 | 112.7 | 128.8 | 144.9 


TABLE XXTV.—lLOGARITHMS OF NUMBERS. 


No. 270 L. 431.] [No. 299 L. 476. | 


N.| 0 1 2 3 4 || 6 6 7 8 9 | Lift. 


| 431364 | 1505 | 1685 | 1846 | 2007 


2167 | 2328 | 2488 | 2649 | 2809 | 161 


270 | 
1 2969 | 81380 } 8290 | 3450 | 3610 || 3770 | 3930 | 4090 | 4249 | 4409 160 
2 4569 | 4729 1 4888 | 5048 | 5207 || 53867 | 5526 | 5685 | 5844 | 6004 159 
3 6163 ‘ 6322 | 6481 | 6640 | 6799 || 6957 | 7116 | 7275 | 7483. | 7592 159 
4 7751 | 7909 | 8067 ' 8226 | 83884 || 8542 | 8701 | 8859. | 9017 | 9175 158 


9333 | 9491 | 9648 | 9806 | 9964 |' —-— |- 
} 0122 | 0279-| 0487 | 0594 | 0752 


1695. | 1852 | 2009 | 2166 | 2823 


SP EE ay ae 15 
440909 | 1066 | 1224 | 1381 15 
2480 | 2687 | 2793 | 2950 | 8106 || 8268 | 3419 | 38576 | 3732 | 8889 | 15 
4045 | 4201 | 4857 | 4513 4669 || 4825 | 4981 | 5137 | 5293 | 5449 | 15 
15 

15 


1588 


5604 | 5760 | 5915 | 6071 | 6226 || 6382 | 6537 | 6692 | 6848 | 7003 
| 158 | 7313 | 7468 | 7623 | 7778 || 7938 | 8088 | 8242 | 8397 | 8552 
706 | 8861 | 9015 | 9170 | 9324 || 9478 | 9633 | 9787 | 9941 


— — =| OOOR at 
450249 | 0403 | 0557 | 0711 | 0865 || 1018 | 1172 | 1326 | 1479 | 1633 | 154 
1786 | 1940 | 2093 | 2247 | 2400 || 2553 | 2706 | 2859 | 3012 | 3165 | 15 
3318 | 3471 | 8624 , 8777 | 3930 || 4082 | 4235 | 4387 | 4540 | 4692 | 153 
| 4845 | 4997 | 5150 | 5302 | 5454 || 5606 | 5758 | 5910 | 6062 | 6214 152 
6366 | 6518 | 6670 | 6821 | 6973 || 7125 | 7276 | 7428 | 7579 | 7731 152 
7882 | 8033 | 8184 | 8336 | 8487 || 8638 | 8789 | 8940 | 9091 | $242 | 151 
9392 | 9543 | 9694 | 9845 | 9995 


pa St 0146 | 0296 | 0447 | 0597 | 0748 | 151 
460898 | 1048 | 1198 | 1348 | 1499 | 1649 | 1799 | 1948 | 2098 | 2248 | 150 


ORWDOE DS OW DODNIOOUIRWMO HO OBOMNS 


29 2398 | 2548 | 2697 | 2847 | 2997 || 8146 | 8296 | 3445 | 3594 44 150 
8893 | 4042 | 4191 | 4840 | 4490 || 4639 | 4788 | 4986 | 5085 | 5234 149 
53883 | 5582 | 5680: | 5829 | 5977 || 6126 | 6274-| 6423 | 6571 | 6719 149 
6868 | 7016 | 7164 | 7312 | 7460 |! 7608 | 7756 | 7904 | 8052 | 8200 148 
8847 | 8495 | 8648 | 8790 | 8988 7; 9085 ; 92383 | 9380 | 9527 | 9675 148 
9822 | 9969 . |. | 
es ——/ 0116 | 02638 | 0410 0557 | O704 | 0851 | 0998 | 1145 147% 
6 71292 | 1488 | 1585 | 1732 | 1878 2025 | 2171 | 2318 | 2464 | 2610 146 
df 2756 | 2903 | 8049 | 3195 | 3841 8487 | 3633 | 8776 8925 | 4071 146 
8 4216 | 4862 | 4508 | 46538 | 4799 4944 | 5090 | 5285 | 5881 | 5526 146 
9 5671 | 5816 | 5962 | 6107 | 6252 397 | GE42 | C687 | 6882 | 6976 145 
PROPORTIONAL PARTS. | 
f 
Diff 1 2 | 3 4 5 6 i 8 9 
161 | 16.1} 32.2 | 48.8 | 644 | 80.5 | 96.6. | 112.7] 128.8 | 144.9 | 
160 | 16.0 | 32.0 | 48.0 | 64.0 | 80.0 | 96.0 | 112.0] 128.0 | 144.0 
159 15.9 31.8 Movil 63.6 79.5 95 .4. 111.3 127.2 | 1438.1 
158 | 15.8 31.6 ad 63.2 79.0 94.8 110.6 126.4; | 142.2 
157 151.7 31.4 {ion 62.8 78.5 94.2 109.9 125.6 | 141.38 
156. «| 15.6 Slee 46.8 62.4 78.0 93.6 109.2 124.8 | 140.4 
155 15.5 31.0 46.5 62.0 (G5) 93.0 108.5 124.0.) 1389.5 
154 15.4. 30.8 46.2 61.6 (00 92.4 107.8 123.2 | 138.6 | 
i laser ira Rages’ 80.6 45.9 61.2 G55 91.8 107.1 122.45) 1st er 
152.-| 15.2 | 380.4 45.6 60.8 76.0 91.2 106.4 121.6 | 136.8 
iSiay) 15.291 SOre 45.3 60.4 RD .5 90.6 105.7 120.8-| 185.9 
150 15.0 + 30:0 45.0 60.0 75.0 90.0 105.0 120.0 | 185.0 
149 | 14.9} 29.8 44.7 59.6 44.5 89.4 104.3 410 Se) Sarr 
148 14.8 29.6 44.4 59.2 .| %.0 88.8 103.6 118.4°) 133.2 
147 14.70) 29.4 44.1 58.8 73.5 88.2 102.9 117.67) A3sa3 
146 14,62) 29%2 43.8 58.4 fo: 0 7.6 oh T0269 116.8 | 131.4 
145 14.5 | 29.0 43.5 58.0 425 87.0 oh 10165 116.0 | 180.5 
144 14.4 28.8 43.2 57.6 72.0 86.4 100.8 115.24 12946 
143 ie Os) 28 .6 42.9 57.2 1.5 85-8 ah 10031 114.4 | 128.7 
142. | 14.2 28.4 42. ¢ 56.8 71.0 85.2 99.4 113.6) 127.8 
144. 4 14.1 28.2 42.3 56.4 70.5 84.6 98.7. | 112.8 | 126.9 
140 | 14.0 | 28.0 2.0 56.0 70.0 84.0 98.0 112.0.) 426.0 


ou) 
awe 
a 


| No. 300 L. 477.) 


TABLE XXIV.—LOGARITHMS OF NUMBERS. 


[No. 339 L. 581. 


0099 | 


N.| 0 | 1 2/ sila | 5 | 6 | 7 | 8 | 9 | Diff. 
|| | | | 

800 : 477121 | 7266 | 7411 | 7555 | 7700 || 7844 | 7989 | 8138 | 8278 | 8422 | 145 
1 8566 | 8711 | 8855 | 8999 | 9143 i 9287 | 943 | 9575 | 9719 | 9863 | 144 

| | a | | 

| 2) 480007 | 0151 | 0294 | 0438 | 0582 |! 0725 | 0869 | 1012 | 1156 | 1289 | 144 

as 1443 | 1586 | 1729 | 1872 | 2016 || 2159 | 2802 | 2445 | 2588 2731 143 

| 4 29874 | 3016 | 38159 | 8802 | 2445 || 8587 | 87380 | S872 | 4015 | 4157 148 

ie ey 4300 | 4442 | 4585 | 4727 | 4869 || 5011 | 5158 | E295 437 | 5579 | 142 
6 5721_-| 5863 | 6005 | 6147 | 6289 || 6480 | 6572 |- 6714 6597 142 
7 7138 | 7280 | 7421 | 7563 | “704 || 7845 | 7986 | 8127 | 8410 | 141 
8 | 8551 |. 8692 | 8833 | 8474 | 9114 9255 | 9896 | 9587 9818 | 141 
9| 9958 | =| | 


9203 


9337 


947 


9606 


| 510545 

1883 
| 38218 
' | 4548 
5874 
7196 


UW 
BHO OMNI C9 


oe 8514 
| 9828 | 
2 | 5211388 
3| 2444 
4). 3746 
‘ae 5045 
6 | 633! 
fg 7630 | 
8 | 8917 


0679 
2017 
38351 
4681 
6006 
7328 
8646 
9959 


1269 
2575 


| 3876 
5 | 5174 | 
| 6469 | 


WG59 


| 9045 


| 0009 


0143 | 


0411 124 
1 


| | 0239 | 0880 | 0520 || 0661 801 | 6841 | 1081 | 1222 | 140 
| 

310 | 491362 | 1502 | 1642 | 1782 | 1922 2062 | 2201 | 2341 | 2481 | 262 140 
1 2760 | 2900 | 3040 | 8179 | 3319 || 3458 | 8597 | 5787 | 8676 | 4015 139 
9) 4155 | 4294 | 4438 | 4572 | 4711 || 4850 | 4989 | 5128 | 5267 | 5406 | 189 
3 | 5544 | 5683 | 5822 | 5960 | 6099 || 6238 | 63876 | 6515 €€58 | 6791 189 
4 6930 | 7068 |.7206 | 7344 | 7483 || 7621 | 7759 | T8S7 | EC25 | 8173 | 188 
5 8311 | 8448 | 8586 | 8724 | 8862 || 8999 | 9187 | S275 | 412 | CEEO | 188 

6 9687 | 9824 | 9962 If _—. = 
| | 0099 | 0286 || 0874 | 0511 | 0648 | OCES | 0922 | 187 
% | 501059 | 1196 | 1333 | 1476 | 1607 | 1744 | 18€0 | £017 | S1E4 |} 2291 1G 
8 | 2497 | 9564 | 2700 | 2887 | 2973 || 8109 | 3246 | 8882 | E518 | E655 126 
9 3791 | 3927-| 4063 | 4199 | 4335 || 4471 | 4607 | 4743 | 46.8 | £014 | 1286 
320 | 5150 | 5086 | 5421 | 5557 | 5693 || 5628 | 5964 | 6099 | 6224 | E870 | 186 
| 6505 | 6640 | 6776 | 6911 | 7046 || 7181 | 7316 | 7451 | EEG | 7721 135 
#256 | 7991 | 8126 | 8260 | 8895 || 8530 | 8664 | 8799 | E924 | C068 | 185 


0813 | 0947 | 1081 || 1215 | 13849 | 1482 | 1616 750 8 
2151 | 2284 | 2418 || 2551 | 2684 | 2818 | 2051 | ced 18: 
3484 | 3617 | 8750 || 8883 | 4016 | 4149 | 42€2 | 4415 182 
4818 | 4946 | 5079 || 5211 | 5844 | 5476 | ECCO | 5741 183 
6139 | 6271 | 6403 || 5 | 6668 | GECO | CGe2 | 764 132 
7460 | 7592 | 7724 || 7987 | 8119 | $251 | G82 | 182 
8777 | 8909 | 9040 | 9171 | £303 | 9484 | S5C6 | SEO" 131 


0090 
1400 
9705 
4006. | 
53804 

6598 
7888 


0221 | 0358 |) 0484 

1530 | 1661 || 1792 | 1922 
2835 | 2966 || 8096 | 3226 
4136 | 4266 || 4896 | 4526 
5434 | 5563 || 5693 | 5822 


6727 | 6856 || 6985 | 7114 
8016 | 8145 || 8274 | 8402 
| 9480 || 9559 | 9687 


0615 


| QOF 


| 9815 | 


0745 
2053 
uve 

4€E6 
5951 
(243 


631 


47&5 | 4916 180 
€c81 | 6210 129 
Gate NBO aes 
FECO | S788 129 
GC 


|. 580200 | 0328 


sy: 
At 


2 || O€ 


PROPORTIONAL PARTS. 


968 


1C96 | 


43 
0072 128 
223 | 1351 128 


39 
a0 | ) 
38 8 
| Qn rn 
af ( 


| MCVBwwwwwwwnw 


Ten os 


Pm be pk Re tek ek ek eek Re ek peek ek PL 


OO OO CO GO CO CO GH C9 GO GO CO GO 


IDOOrWO KE OL 


ee ee 


; MWOOrwwror 


\ 
| 
| 
| 


27.8 
27.6 
27 4 
Ohoe 
27.0 
26.8 
26.6 
26.4 
26.2 
26.0 
95.8 
Ro .6 
25.4 


3 4 5 6 
ret 6 69.5 83 .4 
41.4 £3 69.0 32 .£ 
41-1 8 68.5 82.2 
40.8 | A 68.0 81.6 
40.5 0 67.5 81.0 

ORS D tI 6 67.0 | 80.4 
| 39.9 | 2 oh 66.5. 7958 
} 39.6 | 8 66.0 79.2 
39.3 2.4 65.5 [8.6 
| 89.0 0 65.0 "8.0 
38.7 1.6 64.5 } 77.4 
38.4 2 64.0 76.8 
| 38.1 VE 63.5 76.2 


WAWSEVBAHDEUWONDW 


a pew 
S > 
= Or 
OoMo 


— 
oo) 
rus) 


111.2 | 195-4 
110.4 | 124-2 
109.6 | 123.3 
108.8 | 1£2.4 
108.0 | 121.5 
107.2 | 120.6 
106.4 | 119.47 


~ Aa N-I CS 


G9 OH 


TABLE XXIV.—LOGARITHMS OF NUMBERS, 


=a 


| { No. 840 L, 881.] [No. 879 1. 579. 
1 oe = ===> 


. | 
N.| 0 1 21a) al 51 8 |e 8 | 9 | pitr. 


340 | 531479 | 1607 | 1734 | 128 
1 2754 | 2882 | 3009 | 3136 | 3264 3391 | 8518 | 3645 | 3772 | 3899 | 127 
R 4026 | 4153 | 4280 | 4407 | 4534 |! 4661 | 4787 | 4914 | 5041 | 5167 127 
3 5294 | 5421 | 5547 | 5674 | 5800 || 5927 | 6053 | 6180 | 6306 | 6432 126 


1862 | 1990 || 2117 | 2245 | 2372 


2500 | 2627 


4 6558 | 6685 | 6811 | 6937 | 7063 || 7189 | 7315 : 7441 | 7567. | 7693 126 
5 7819 | 7945 | 8071 | 8197 | 8322 || 8448 | 8574 | 8699 | 8825 | 8951 
6 
7 


9076 | 9202 | 9327 | 9452 | 9578 || 9703 | 9829 | 9954 a 
| ———,— —| 0079 | 0204 | 125 
640329 | 0455 | 0580 | 0705 | 0830 || 0955 | 1080 | 1205 | 1330 | 1454 | os 
8; 1579 | 1704 | 1829 | 1953 | 2078 | 2203 | 2827 | 2152 | 2576 | aroq | 
9] 2825 | 2950. | 8074 | 8199 | 3323 3447 | 3571 | 8696 | 3820 | 3944 | jo4 
350 | 4068 | 4192 | 4316 | 4440 | 4564 | 4688 | 4812 | 4936 | 5060 | 5188 | 124 
Wn 1 530% | 5431 | 5555 | 5678 | 5802 | 5925 | 6049 | 6172 | 6296 | E419 | 2 
Ne i 2 | 6543 | 6666 | 6789 | 6913 | 7036 || 7159 | 7282 | 7405 | 7529 | 7659 | jO3 
ct 3 7¢S | 7898 | 8021 | 8144 | 8267 | 8389 | 8512 | 8635 | 8758 | 8881 23 
4 9003 | 9126 | 9249 | 9371 | 9494 | 9616 | 9739 | 9861 | 9984 | 
gpa = : | 0106 | 128 
5 | 550228 | 0351 | 0473 | 0595 | 0717 | 0840 | 0962 1084 | 1206 | 1328 | 122 
6 1450 | 1572 | 1694 | 1816 | 1938 | 2060 | 2181 | 2303 | a49% 2547 | 122 
7 
8 
9 


2668 | 2790 | 2911 | 8033 | 3155 || 8276 | 3398 | 3519 | 3640 | 2g | Jo 


8883 | 4004 | 4126 | 4247 | 4368 || 4489 | 4610 | 4731 | 4959 | 4973 | 121 


5094 | 5215 | 5336 | 5457 | 5578 | 569 | 5820 | 5940 | Go61 | Gige | jot 
360 | 6303 | 6423 | 6544 | 6664 | 6785 |/ 6905 | 7026 | 7146 | 7267 | e397 | 490 
Py) 7507 | 7627 | 7748 | 7868 | 7988 || 8108 | 8228 | 8349 | g169 | gaeq | 490 
2) 8709 | 8829 | 8948 | 9068 | 9188 || 9308 | 9428 | 9548 | 9667 | 9787 | 120 
3] 9907 | —— | 
~——~| 0026.| 0146. | 0265 | 0385 | 0504 | 0624 | 0743 | 0863 | cose | 119 
SULL0L | 1221 ) 1340 | 1459 | 1578 || 1698 | 1817 | 1936 | 2055 | oye | 179 
2293 | 2412 | 2531 | 2650 | 2769 8006 | 3125 | 3244 | 3362 | 119 
8481 | 3600 | 8718 | 8837 | 3955 || 4074 | 4192 | 4311 | 4429 | gs4g | 119 
4666 | 4784 | 4903 | 5021 | 5139 || 5257 | 5376 | 5494 | 5612 | 5730 | 148 
5848 | 5966 | 6084 | 6202 | 6320 || 6437 | 6555 | 6673 | 6791 | 6909 | 118 
W026 | 144 | 7262 | 7379 | 7497 || 7614 | 7732 | 7849 | 7967 | gos, | 118 
8202 | 8319 | 8436 | 8554 | 8671 || 8788 | 8905 | 9028 | 9140 | gos | 447 
9374 | 9491 | 9608 | 9725 | 9842 |} 9959 a 
. | = —| 0076 | 0193 | 0309 | 0426 | 4117 
570543 | 0680 | 0776 | 0893 | 1010 |, 1126 | 1243 | 1359 | 1476 | 1599 | 117 
1109 | 1825 | 1942 | 2058 | 2174 || 2291 | 2407 | 2523 | 2639 | aras | 146 
2872 | 2988 | 3104 | 3220 | 3336 || 3568 | 3684 | 3800 | 3915 | 116 
A031 | 4147 | 4263 | 4879 | 4494 || 4610 | 4726 | 4841 | 4957 | 5072} 116 
5188 | 5303 | 5419 | 5534 | 5650 || 5765 | 5880 | 5996 | 6111 | 6296 | 115 
Gaal | 6457 | 6572 | 6687 | 6802 || 6917 | 7032 | 7147 | 7262 | 3771 715 
(492 | 7607 | 7722 | 7836 | 7951 || 8066 | 8181 | 8295 | 8110 | 85951 i7s 
8639 | 8754 | 8868 | 8983 | 9097 | 9212 | 9326 | 9441 | 9555 | 9¢60 | 14 


2» 
@ 
oa 
NS 


is) 


COImOMAwWs HS CMOMOOR 


Oo 
NS 
Or 
rw) 


es 
Qa 
on 
ror 


PROPORTIONAL Parts. 
| — 
ee Oe ae oh a ge! 7 8 9 
| | | 
5 Sam | / | aD ae Moy Ts 
128 | 12.8] 25.6 | 38.4 51.2 | 64.0 “6.8 | 89.6 | 102 115.2 
127 | 12.7] 25.4 B81) 50.8 63.5 76.2 | 88.9 | 101.6 | 114.3 
126 | 12.6] 25.2 37.8 | 50.4 63.0 15.6 “| 88/2 2100-8 418.4 
125 | 12.5 | 25.0 87.5 | 50.0 | 62.5 75.0 | 87.5 | 100.0 | 112.5 
12 12.4] 24.8 37.2 | 49.6 | 62.0 74.4 | 86.8 99.2 | 111.6 
. 123- | 12.3} 24.6 36.9: |. 49.9 \) 61.5 73:8 | 86.1 98.4 -| 110.7 
eth Iie) 2440+ 26.6 | 48.8 | 61.0 73.2 | 85.4 | 97.6 | 109.8 
ane) 124 24.0% 368 8.4 60.5 (2.6 | 84.7 | 96.8 | 108.9 
120 12.0 24.0 | 36.0 48.0 60.0 72.0 84.0 96.0 | 108.0 
| TGS) TL 23.84 35M ape 59.5 71.4 


| 
| 
| 
| 
| 


83.5 | 95.9 LO tele 


TABLE XXIV.—LOGARITHMS 


OF NUMBERS. 


aye 


No. 380. L. 


9.] 


[No. 414 L. 617. 


| 


6 


So 7.9 |) Der 


579784 | 9898 | ~ a) = 
SOS ee | 0S. 0355 0469 | 0583 | 0697 | 0811 | 114 
1 580925 | 1039 1153 | 1495 | 1608 | 1722 336 | 1950 
2} 2063 | 2177 | 2291 2631 | 2745 | 2858 | 2972 | 3085 
3 3199 | 3312 | 3426 | § 3765 | 8879 | 3992 | 4105 | 4218 
4} 4331 | 4444 | 4557 | 4896 | 5009 | 5122 | 5235 | 5348 | 113 
5 | 5461 | 5574 | 5686 | 6024 | 6137 | 6250 | 6362 | 6475 
6 6587 | 6700 | 6812 | 7149 | 7262 | 7374 | 7486 | 7596 
7 G711 | 7823 | 7935 | || 8272 | 8384 | 8496 | 8608 | 8720 | 112 
8 8832 | 8944 | 9056 || 9391 | 9503 | 9615 | 9726 | 9836 
9 9950 | | | 
— | 0061 | 0173 0507 | 0619 | 0730 | 0842 | 0953 
9 | 591065 | 1176 | 1287 | 1621 | 1732 | 1843 | 1955 | 2066 
1} 2177 | 2288 | 2399 | 2732 | 2843 | 2954 | 8064 | 3175 | 111 
2} 3286 | 3397 | 3508 3840 | 3950 | 4061 | 4171 | 4282 
3 | 4893 | 4503 | 4614 | 4945 | 5055 | 5165 | 5276 | 5386 
4| 5496 | 5606 | 5717 | 6047 | 6157 | 6267 | 6577 | 6487 
5 | 6597 | 6707 | 6817 | "146 | 7256 | 7366 | 7476 | 7586 | 110 
6 "695 | 7805 | 7914 8243 | 8353 | 8462 | 8572 | 8681 
4 8791 | 8900 | 9009 9337 | 9446 | 9556 | 9665 | 9774 
8 9883 | 9992 | — 109 
Si 2 eee Hotel 0428 | 0537 | 0646 | 0755 | 0864 
9 | 600973 | 1082 | 1191 1517 | 1625 | 1734 | 1843 | 1951 
() 2060 | 2169 | 2277 | § | 2603 | 2711 | 2819 | 2928 | 3036 
1 3144 | 3253 | 2361 3686 | 8794 | 8902 | 4010 | 4118 | 408 
Q 4226 | 4334 | 4442 | | | 4766 | 4874 | 4982 | 5089 | 5197 
3 | 5305 | 5413 | 5521 | | 5844 | 5951 | 6059 | 6166 | 6274 
4 6381 | 6429 | 6596 | 6919 | 7026 | 7133 | 7241 | 7348 
5 7455 | 7562 | 7669 | || 7991 | 8098 | 8205 | 8312 | 8419 | 407 
6} 826 | 8633 | 8740 | 9061 | 9167 | 9274 | 9381 | 9488 
“i 9594 | 9701 | 9808 | eS 
ee eats 0128 | 0234 | 0341 | 0447 | 0554 
8 | 610660 | 0767 | 0873 | |} 1192 | 1298 | 1405 | 1511 | 1617 
1723 | 1829 | 1936 | 2254 | 2860 | 2466 | 2572 | 2678 | 106 
2784 | 2890 | 2996 8313 | 3419 | 3525 | 3630 | 3736 
8842 | 3947 | 4053 | 4370 | 4475 | 4581 | 4686 | 4792 
4897 | 5003 | 5108 | 5424 | 5529 5634 | 5740 | 5845 
5950 | 6055 | 6160 | 6476 | 6581 | 6686 | 6790 | 6895 | 105 
7000 | 7105 | 7210 "525 | 7629 | 7734 | 7839 | 7943 
PROPORTIONAL PARTS. 
| | 
11.8 | 28.6 | 35.4 “3 59.0 "0.8 82.6 94.4 | 106.2 
10.7 1593.47" 35.1 8 58.5 70.8 81.9 | 93.6 | 105.3 
IG a es 2 of 34-R 4 58.0 69 81.2 | 92.8 | 104.4 
1115. | 23.0 34.5 46.0 | 57.5 69. 80.5 | 92.6 | 1038.5 
| 11.4 |' 22.8 24.2 45.6 | 57.0 68 "9.8 | 91.2 | 102.6 
| 11.34 22.6 33.9 45.2 | 56.5 64 79.1 90.4 | 101.7 
| 11.2] 22.4 | 33.6 | 44.8 | 56.0 | 67 78.4 | 89.6 | 100.8 
| | 
pete ADS. 33.3 44.4 | ° 55.5 66. fay? 88.8 | 99.9 
111.0! 22.0 33.0 44.0 55.0 66. 77.0 88.0 99.0 
10.9 | 21.8 32.7 43.6 54.5 65. "6.3 8732 | 5a 
10.8 | 21.6 32.4 43.2 54.0 64. "5.6 86.4 97.2 
102% | 2.4 32.1 42.8 53.5 64 | AOI) 85.26 96.3 
1028 |) 21.27 | 4 8H8 42.4 53.0 3, 74.2 | 84.8 | 95.4 
10.5] 21.0 | 31.5 42.0 52.5 33.0 | %B.5 84.0 94.5 
10.5.| 21.0 | 31.5 42.0 52.5 63.0 | 73.5 84.0 94.5 
10.4| 20.8 | 31.2 | 41.6 | 52.0 72.8 | 83.2 | 93.6 


TABLE XXIV.—LOGARITHMS 


OF NUMBERS. 


{ 
| 
| 


No, 415 L, 618.] [No. 459 L. 662 | 
1| 
N.| 0 122% | 4 || 56 | 6 ro s Diff. 
| | | 
| | | | | | 
415 | 618048 | 8153 | 857 | 8362 | 8466 || 8571 | 8676 | 8780 | gas4 | 8989 || 105 
6 | 9093 | 9198 | 9302 | 9406 | 9511 || 9615 | 9719 | 9824 | 9928 |-——— 
——- 0082 | 
7 | 620136 | 0240 | 0344 | 0448 | 0552 || 0656 | 0760 | 0864 | 0968 | 1072 | 104 
8 | 1176 | 1280 | 1384 | 1488 | 1592 || 1695 | 1799 | 1903 | 2C07 | 2110 
9 | 2214 | 2318 | 2421 | 2525 | 2628 || 2732 | 2835 | 2939 | 3042 | 3146 
420 3249 | 3353 | 3456 | 8559 | 3663 || 38766 | 3869 | 3973 | 4076 | 4179 
1 | 4282 | 4385 | 4488 | 4591 | 4695 || 4798 | 4901 | 5004 | 5107 | 5210 | 103 
2 | 5812 | 5415 | 5518 | 5621 | 5724 || 5827 | 5929 | 6082 | 6185 | 6238 
3 | 6340 | 6443 | 6546 | 6648 | 6751 || 6853 | 6956 | 7058 | 7161 | 7263 
4 7366 | 7468 | 7571 | 7673 | 7775 || 7878 | 7980 8082 | 8185 | 8287 
i 5 | 83889 | 8491 | 8593 | 8695 | 8797 || 8900 | 9002 | 9104 | 9206 | 9308 | 402 
6 | 9410 | 9512 | 9613 | 9715 | 9817 || 9919 ; 
| | 0021 | 0123 | 0224 | 0526 
% | 630428 | 0530 | 0631 | 0733 | 0835 || 0936 | 1088 | 1189 | 1241 | 1342 | 
8 1444 | 1545 | 1647 | 1748 | 1849 || 1951 | 2052 | 2158 | 2255 | 2356 | 
i 9 | 2457 | 2559 | 2660 | 2761 | 2862 |) 2963 | 8064 | 3165 | 8266 | 3867 | 
Hi 430 | 3468 | 3569 | 38670 | 3771 | 3872 |) 8973 | 4074 | 4175 | 4276 | 4376 | 101 
i | 1 4477 | 4578 | 4679 | 4779 | 4880 || 4981 | 5081 | 5182 | 5283 | 5388 
2| 5484 | 5584 | 5685 | 5785 | 5886 || 5986 | 6087 | 6187 | 6287 | 6388 
3 | 6488 | 6588 | 6688 | 6789 | G889 || E9&9 | 7089 | 7189 | 7290 | 7390 | 
4 | 7490 | 7590 | 7690 | 7790 | 7200 || 7990 | 8C90 | 8190 | 8290 | 83889 | 499 
5 | 8489 | 8589 | 8689 | 8789 | 8888 || 8988 | 9088 | 9188 | 9287 | 9387 
6 | 9486 | 9586 | 9686 | 9785 | 9885 || 9984 + _| 
— | | 0084 | 0183 | 0283 | 0382 
7 | 640481 | 0581 | 0680 oe | C879 |; 0978 | 1077 | 1177 | 1276 | 1875 
8 1474 | 1573 | 1672 | 1771 | 1871 || 1970 | 2069 | 2168 | 2267 | 2366 
9 | 2465 | 2563 | 2662 | 2761 | 2s6v | 2059 | 3058 | 3156 | 8255 | 3854 99 
440 | 3453 | 8551 | 3650 | 3749 | 8847 || 3946 | 4044 | 4143 | 4242 | 4340 
1 | 4489 | 4587 | 4636 | 4734 | 4282 || 4931 | 5029 | 5127 | 5226 | 5824 
2 | 5422 | 5521 | 5619 | 5717 | 5815 || 5913 | 6011 | 6110 | 6208 | 6806 
3 | 6404 | 6502 | 6600 | 6698 | 6796 || 6804 | 6992 | 7089 | 7187 | 7285 98 
4) 4 | 7883 | 7481 | 7579 | 7676 | 7774 || 7872 | 7969 | 8067 | 8165 | 8262 
5 | 8860 | 8458 | 8555 | 8653 | 8750 || 8848 | 8945 | 9043 | 9140 | 9287 
6 | 9335 | 9432 | 9530 | 9627 | 9724 || 9821 | 9919 i 
_——— | | 0016 | 0113 | 0210 
7 | 650308 | 0405 | 0502 | 0599 | 0696 |) 0793 | 0890 | 0987 | 1084 | 1181 2 
8} 1278 | 1375 | 1472 | 1569 | 1666 || 1762 | 1859 | 1956 | 2053 | 2150 97 
9 | 2246 | 2348 | 2440 | 2536 | 2633 | 2730 | 2826 | 2923 | 3019 | 3116 
| 450 | 3213 | 3809 | 3405 | 3502 | 85598 || 2695 | 8791 | 3888 | 3984 | 4080 
| 1 4177 | 4273 | 4369 | 4465 | 4562 || 4658 | 4754 | 4850 | 4946 | 5042 
2} 5138 | 5235 | 5331 | 5427 | 5523 || 5619 | 5715 | 5810 | 5906 | 6002 | 96 
3 | 6098 | 6194 | 6290 | 6386 | 6482 || 6577 | 6673 | 6769 | 6864 | 6960 
4| 7056 | 7152 | 7247 | 7343 | 7488 || 7534 | 7629 | 7725 | 7820 | 7916 
5 | 8011 | 8107 | 8202 | 8298 | 8393 || 8488 | 8584 | 8679 | 8774 | S870 
hie 6 | 8965 | 9060 | 9155 | 9250 | 9346 || 9441 | 9536 | 9631 9726 | 9821 
HT 7 | 9916 | | 
i} ————} 0011 | 0106 | 0201 | 0296 || c391 | 0486 |/0581 | o676 | O77 95 
i 8 | 660865 | 0960 | 1055 | 1150 | 1245 |) 1339 | 1434 | 1529 | 1623 | 1718 
} 9 1813 | 1907 | 2002 | 2096 | 2191 |r 2286 | 2380 | 2475 | 2569 | 2663 
iil PROPORTION AL Parts. 
| Diff.| 1 2 3 4 5 6 7 8 9 
105) 10.5! )- 21:0 | 88181) 42.0], 52.52 15 68.0) | 738.5 1 S41 pees 
104 | 10.4] 20.8 31.2 41.6 52.0 62.4 72 8 83.2 93.6 
103.1103) |-5.20:6 | 80:98). 42) SRB] 6h Be) F2a4 | “Roa a oes 
102 | 10.2 | 20.4 | 30.6 | 40.8 | 51.0 | 61.2 | 71.4 | 81.6 91.8 
101 | 10.1] 20.2 30.3 40.4 50.5 60.6 707 | 
{ 100 | 10.0] 20.0 30.0 40.0 50.0 60.0 70 0 
89h OD (19: 8)> | 296% 1.6 | 49.5 | 59.4 | 69.3 


TABLE XXIV.—LOGARITHMS OF NUMBERS. 


| 


I No, 460 L. 662.1 


Ate 
. 499 L. 698. | 


| i 
N | 0O 2 8 4 5 @ 8 Diff. 
= ‘ ah | 3 = ———— 
460 | 662758 | 2947 | 3041 | 3135 || 3220 3418 | 3512 
1 3701 3889 | 8983 | 4078 ||. 4172 | 4360 | 4454 
2| 4642 | 4830 | 4924 | 5018 || 5112 | 5299 | 5393 94 
8 | 5581 5769 | 5862 | 5956 || 6050 | 6143 | 6287 | 6331 
4 6518 | 6705 | 6799 | 6892 || 6986 | 7079 | 7173 | 7266 
5 7453 7640 | 7733 | 7826 || 7920 | 8013 | 8106 | 8199 | 
6 8386 | 8572 | 8665 | 8759 || 8852 | 9038 | 9131 
‘i 9317 | 9503 | 9596 | 9689 || 9782 | 9875 | 9967 |— 
: ie iy | - 0060 | 98 
8 | 670246 | 0431 | 0524 | 0617 || 0710 | 0895 | 0988 
9 1173 | | 1358 | 1451 | 1543 || 1636 | 1728 | 1821 | 1918 
470 | 2098 | 2283 | 2375 | 2467 || 2560 2 | 2744 | 2836 
1} 3021 | | 3205 | 3297 | 3390 || 3482 3666 | 3758 
2 3942 4126 | 4218 | 4310 |) 4402 4586 | 4677 | 92 
3 | 4861 | 5045 | 5137 | 5228 || 5820 5508 | 5595 | 
4| 5778 | | 5962 | 6053 | 6145 | 6236 6419 | 6511 
5 | 6694 | | 6876 | 6968 | 7059 | 7151 | 7242 | 7388 | W424 | 
6 | 7607 7789 | 7881 | 7972 || 8063 | 8154 | 8245 | 8336 
| 7] 8518 | 8700 | 8791 | 8882 | 8973 | 9064 | 9155 | 9246 91 
8 | 9428 | | 9610 | 9700 | 9791 | 9882 | 9973 | - | 
| ee | 0063 | 0154 
9 | 680336 | 0517 | 0607 | 0698 || 0789 | 0879 | 0970 | 1060 | 
480 | 1241 | | 1422 | 1513 | 1603 || 1693 1874 | 1964 
1 2145 | % 2326 | 2416 | 2506 || 2596 | 2686 | 2777 | 2867 
2| 3047 37 | 8227 | 8817 | 3407 || 3497 3677 | 3767 90 
8} 3947 | | 4127) 4217 | 4307 || 4396 4576 | 4666 
4 | 4845 | 5025 | 5114 | 5204 | 5294 | 5473 | 5563 
5 | 5742 | 5831 | 5921 | 6010 | 6100 || 6189 | 6368 | 6458 
6 6636 6815 | 6904 | 6994 | 7083 | 7261 | 7851 
Earle 7529 | 7618 | 7707 | 7796 | 7886 | 7975 | 8064 | 8153 | 8242 89 
| 8 |) 8420 | 8509 | 8598 | 8687 | 8776 | 8865 | 8953 | 9042 | 9181 
Big 9309 | 9486 | 9575 | 9664 | 9753 | | 9930. | 
— | | ——) 0019 
490 | 690196 0373 | 0462 | 0550 | 0639 | 0728 | 0816 | 0905 
1 1031 | 1258 | 1847 | 1435 || 1524 | 1700 | 1789 
2 1965 | 2142 | 2280 | 2318 || 2406 | 2583 | 2671 
3 | 2847 | 3023 | 3111 | 3199 || 3287 3463, | 3551 88 
| 4 | 3727 | 3903 | 3991 | 4078 || 4166 4342 | 4430 
5 | 4605 4731 | 4868 | 4956 || 5044 | 5219 | 5307 | 
6 5482 | | 5657 |. 5744 | 5882 || 5919 6094 | 6182 
7 | 6356 | 6531 | 6618 | 6706 || 6793 6968 | 7055 | 
8 7229 | 7404 | 7491 | 7578 || 7665 7839 | 7926 
9 | 8100 | 8275 | 8362 | 8449 || 8535 | 8709 | 8796 87 
| } 
PROPORTIONAL PARTS. 
| 
lela] 2 fs] 4s 
98 | 9.1 | 29.4 | 80.2 | 49.0 | 58.8 | 68.6 8. 88.2 | 
i ae ey 29.1 38.8 48.5 58.2 67.9 Wop 87.3 | 
96 *; 9.6 | 28.8 38.4 48.0 57.6 67.2 6. 86.4 | 
95. | 9.5 28.5 38.0.| 47.5 57.0 | 66.5 6 85.5 
94 9 23.2 37.6 47.0 56.4 65.8 $4.0 | 
, 93 9.3 L279 37.2 -|—46.5 55.8 65.1 83.4 
92 9.2 | 27.6 | 36.8 46.0 55.2 64.4 82.8 
91 | 9.1 Pee 8) Ao. 4 ous 4o.b 54.6 63.7 81.9 
90 | 9. | 27.0'-| 86.0 | 45.0 | 54.0 | 63.0 81.0 
89 | 8.9 | 26.7 | 35.6 | 44.5 53.4 32.3 80.1 
88 8.8 26.4 | 35.2 44.0 52.8 61.6 79.2 
| | 94:8 3.5 | 52.2 60.9 78-3 
\evd4ia 0 51.6 f 0% 4 


| & 
vo 


TABLE XXIV.—LOGARITHMS OF NUMBERS. 


: 


No, 500 L. 698.] [No. 544 L, 736. 


or 


Nala 0% |) ig | 8 | | aed 6 | 7 | 8 | 9 | pi. 
| | | 


500 | 698970 | 9057 | 9144 | 9231 | 9817 || 9404 | 9491 | 957 | 9664-| 9751 
1 | 9888 | 9924 | | | | _| | 
| 0011 | 0098 | 0184 | 0271 | 0358 | 0444 | 0531 | 0617 | 


(0 6) 


2 | 700704 | 0790 | 0877 | 0963 | 1050 || 1136 | 1222 13809 | 1895. | 1482 | 

3} 1568 | 1654 | 1741 | 1827 | 1913 || 1999 | 2086 R172 | 2258 | 2344 | 

4 | 2431 | 2517 | 2603 | 2689 | 2775 || 2861 | 2947 | 3083 | 3119 | 8205 | 

S| 38291 | 3377 | 3463 | 3549 | 3635 || 3721 | 3807 3893 | 3979 | 4065 86 
6 4151 | 4236 | 4822 | 4408 | 4494 || 4579 | 4665 | 4751 | 4837 | 4922 

7 5008 | 5094 | 5179 | 5265 | 5350 || 5436 | 5522 | 5607 | 5693 | 5778 

8 5864 | 5949 | 6035 | 6120 |- 6266 || 6291 | 6376 | 6462 | 6547 | 6632 


9 6718 | 6803 | 6888 | 6974 | 7059 || 7144 | 7229 | 7315 | 7400 | 7485 


HB 510 7570 | 7655 | 7740 | 7826 | T9114 | 7996 | 8081 | 8166 | 8251 | 8336 85 
} 1 8421 | 8506 | 8591 | 8676 8761 || 8846 | 8931 | 9015 | 9100 9185 
i 2 9270 | 9855 | 9440 | 9524 | 9609 || 9694 | 9779 | 9863 | 9948 | — 
| | _|| | 0033 
3 | 710117 | 0202 | 0287 | 0871 | 0456 | 0540 | 0625 | 0710 | 0794 | 0879 
i 4 0963 | 1048 | 1132 | 1217 | 1301 1885 | 1470 | 1554 | 1639 | 1793 
j 5 1807 | 1892 | 1976 | 2060 | 2144 2229 | 2313 | 2397 | 2481 | 2566 
6 2650 | 2734 | 2818 | 2902 | 2986 8070 | 3154 | 8288 | 3323 | 3407 84 
ff 3491 | 3575 | 3659 | 3742 | 3896 | 3910 | 3994 | 4078 | 4162 | 4246 : 
8 4330 | 4414 | 4497 | 4581 | 4665 4749 | 4883 | 4916 | 5000 | 5084 
9 5167 | 5251 | 5335 | 5418 | 5502 || 5586 5669 753 | 5886 | 5920 
520 6003 | 6087 | 61% 6254 | 6337 || 6421 | 6504 | 6588 | 6671 754 
1 6838 | 6921 | 1004 | 7088 | 7171 | 7254 | 73888 | 7421 | 7504 | 7587 
2 7671 | 7754 | 7837 | 7920 | 8003 || 8086 | 8169 | 8253 | 8336 | 8419 83 
3 8502 | 8585 | 8668 | 8751 | 8834 || 8917 | 9000 | 9083 | 9165 | 9248 : 
4 9331 | 9414 | 9497 | 9580 | 9663 || 9745 | 9828 | 9911 | 9994 | 
| | | | 0077 
5 | 720159 | 0242 | 0325 0407 | 0490 || 057% | 0655 | 0788 | 0821 | 0903 
6 0986 | 1068 | 1151 | 123% 1316 || 1898 | 1481 | 1563 | 1646 1728 
ff 1811} 1893 | 1975. | 2058 | 2140 | 2222 | 23805 | 2387 | 2469 | 9559 
8 2634 | 2716 | 2798 | 2881 | 2963 || 8045 | 3127 | 3209 | 3291 | 3974 
9 3456 | 2588 | 3620 | 3702 | 3784 || 3866 | 3948 | 4080 | 4112 | 4194 82 


530 4276 | 4358 | 4440 | 4522 | 4604 || 4685 | 4767 | 4849 4951 | 5013 
5095 | 5176 | 5258 | 5340 | 5422 || 5503 | 5585 | 5667 | 5748 | 5830 
2 5912 | 5993 | 6075 | 6156 | 6238 || 6320 | 6401 | 6483 6564 | 6646 
3 6727 | 6809 | 6890 | 6972 | 7053 || 7134 | 7916 7297 | 7379 | 7460 
4 (41 | 7623 | 7704 | 7785 | 7866 || 7948 | 8029 | 8110 8191 | 827 
5 8354 | 8435 | 8516 | 8597 | 8678 || 8759 | 8841 | So29 9003 | 9084 
6 9165 | 9246 | 9327 | 9408 | £489 || 9570 | 9651 | 9732 .9813 | 9898 81 
7 9974 / 


= 
~) 
S) 


0055 | 0186 | 0217 | 0298 || 0378 | 0459 | 0540 | 062 “02 
8 | 780782 | 0863 | 0944 | 1024 | 1105 || 1186 | 1266 1347 | 1428 | 1508 
9 1589 | 1669 | 1750 | 1830 | 1911 {| 1991 | 2072 | 2152 | 2238 | 2313 


iH 540 | =. 2394 | S474 | 2555 | 2635 | 2715 || 2796 | 2876 | 2956 | 3037 | 3117 
ana To eae 3197 | 8278 | 3358 | 3438 | 8518 || 3598 | 3679 | 3759 | 3889 3919 


2 3999 | 4079 | 4160 | 4240 | 4320 || 4400 | 4480 | 4560 | 4640 | 4720 80 
3 4800 | 4880 | 4960 | 5040 5120 || 5209 | 5279 | 5359 | 5489 | 5519 
Til 4 5599. | 5679 | 5759 | 5838 | 5918 | 5998 | 6078 | 6157 | 6237 | 6317 


PROPORTIONAL PARTS. 


TABLE XXIV.—LOGARITHMS OF NUMBERS. 


r a a a a 
No. 545 L. 786.] LNo, 584 L. 767. 


mi Of Fe paps |e] s pe pa) s | ov pig 


== | 


545 | 736397 | 6476 | 6556 | 6635 | 6715 | 6795 | 6874 | 6954 | 7034 | 7113 | 
| 6) 7193 | 7272 | 7352 | 7431 | 7511 || 7590 | 7670 | 7749 | 7829 | 7908 | 
7987 | 8067 | 8146 | 8225 | 8305 || 8384 | 8463 | 8543 | 8622 | S101 | 
8 8781 | 8860 | 8939 | 9018 | 9097 || 9177 | 9256 | 9335 | 9414 | 9493 | 
9572 | 9651 | 9731 | 9810 | 9889 || 9968 | oe a 
| | 0047 | 0126 | 0205 | 0284} 79 
| 550 740363 | 0442 0521 | 0600 | 0678 || 0757 | 0886 | 0915 | 0994 | 1073 
1 | 11527] 1230 | 1309 | 1388 | 1467 || 1546 | 1624 | 1703 | 1782 | 1860 | 
2) 1939 | 2096 | 2175 | 2254 || 2332 | 2411 | 2489 | 2568 | 2647 | 
3| 2725 | 2804 | 2882 | 2961 | 3039 || 3118 | 3196 | 3275 | 3353 | 3431 iM 
4 3510 | 3588 | 3667 | 3745 | 3823 || 3902 | 38980 | 4058 | 4136 | 4215 t 
5 | 4293 | 4371 | 4449 | 4528 | 4606 || 4684 | 4762 | 4840 | 4919 | 4997 | 
6 | 5075 | 5153 | 5231 | 5809 | 5387 || 5465 | 5543 | 5621 | 5699 | 5777 | 78 | 
7 | 5855") 5933 | 6011 | 6089 | 6167 |) 6245 | 6323 | 6401 | 6479 | 6556 . 
8 | 6634 | 6712 | 6790 | 6868 | 6945 || 7023 | 7101 | 7179 | 7256 | 7334 
9 | 7412 | 7489 | 7567 | 7645 | 7722 || 7800 | 7878 | 7955 | 8038 | 8110 
560 | 8188 | 8266 | 8343 | 8421 | 8498 || 8576 | 8653 | 8731 | 8808 | 8885 
1} 8963 | 9040 | 9118 | 9195 | 9272 || 9350 | 9427 | 9504 | 9582 | 9659 | 
2| 9736 | 9814 | 9891 | 9968 | 


2 
= 
(e 6) 


| ee | 0045 || 0123 | 0200 | 0277 | 0354 | 0431 
3 | 750508 | 0586 | 0663 | 0740 | 0817 || 0894 | 0971 | 1048 | 1125 | 1202 | Hi 
4} 1279 | 1856 | 1433 | 1510 | 1587 |, 1664 | 1741 | 1818 | 1895 | 1972 | os i 
5 2048 | 2125 | 2202 | 2279 | 2356 || 2483 | 2509 | 2586 | 2663 | 2740 ‘ . 
6 | 2816 | 2893 | 2970 | 8047 | 3123 |! 8200 | 8277 | 8853 | 3430 | 3506 { 
7 | 3583 | 3660 | 3736 | 3813 | 8889 |, 8966 | 4042 | 4119 | 4195 | 4272 ; 
8 | 4848 | 4425 | 4501 | 4578 | 4654 || 4730 | 4807 | 4883 | 4960 | 5036 | 
9 5112 | 5189 | 5265 | 5841 | 5417 || 5494 | 5570 | 5646 | 5722 | 5799 i 

570 | 5875 | 5951 | 6027 | 6103 | 6180 |; 6256 | 6882 | 6408 | 6484 | 6560 i 
1 | 6636 | 6712 | 6788 | 6864 | 6940 || 7016 | 7092 | 7168 | 7244 | 7320 16 i 
2| 7896 | 7472 | 7548 | 7624 | 7700 || 7775 | 7851 | 7927 | 8003 | 8079 t 
3 8155 | 8230 | 8306 | 8382 | 8458 || 8533 | 8609 | 8685 | 8761 | 8836 i 
4 8912 | 8988 | 9063 | 9139 | 9214 || 9290 | 9866 | 9441 | 9517 | 9592 i 
5 | 9668 | 9743 | 9819 | 9894 | 9970 | 


| 0045 | 0121 | 0196 | 0272 | 03847 


6 | 760422 | 0498 | 0573 | 0649 | 0724 | 0799 0875 | 0950 | 1025 | 1101 , 
7 | 4176 | 1251 | 1826 | 1402 | 1477 || 1552 | 1627 | 1702 | 1778 | 1853 | 
8 1928 | 2003 | 2078 | 2153 | 2228 |! 2808 | 2878 | 2458 | 2529 | 2604 | a» 
9} 2679 | 2754 |.2829 | 2904 | 2978 || 3053 | 3128 | 3203 | 3278 | 3358, 
580 | 8428 | 3503 | 8578 | 3653 | 3727 || 3802 | 3877 | 3952 | 4027 | 4101 
1| 4176 | 4251 | 4326 | 4400 | 4475 || 4550 | 4624 | 4699 | 4774 | 4848 i 
2 | 4923 | 4998 | 5072 | 5147 | 5221 || 5296 | 5870 | 5445 | 5520 | 5594 l 
3 | 5669 | 5743 | 5818 | 5892 | 5966 || 6041 | 6115 | 6190 | 6264 | 6838 i 
4| 6413 | 6487 | 6562 | 6636 | 6710 || 6785 | 6859 | 6933 | 7007 | 082 H 
| | | 
PROPORTIONAL PARTS, 
pi,| i | 2 3 | 4 5 6 4 8 9 
| | | 
§3 83 16.6 | (24.9 | 33.2 41.5 49.8 58.1 66.4 (4.7 
R2 Spor 16.4. |t-24.Gi al sone 41.0 49.2 57.4 65.6 73.8 | 
81 Ae oy eS SP B24 40.5 48.6 56.4 64.8 2.9 
80 8.0| 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 
Bee Or) 158812) 28; Ar |i" Blues! a9eGen | 47,4 5D. 63.2° | 74:1 
7 | 7.8) 1516 23.4 31.2 | 39.0 46.8 54.6 62.4 (0.2 
me | 7.7 | 15.4 23.1 30.8 | 38.5 46.2 53.9 61.6 | 69.3 
“6 | 7.6-| 15.2 | 22.8 | 30.4 | 388.0 | 45.6 53.2 | 60.8 | 68.4 
fo-75 |. %.5a|. 15.0 22.5. | 30.0 | 37.5 | 45.0 52.5 60.0 v5 
| 7% | v4 | 14.8 22.2 | 29.6 37.0 14.4 51.8 | 59.2 | 66.6 


or 


or 


for) 


62 


TABLE XXTV. 


LOGARITHMS OF NUMBERS. 


8651 | 


8720 


8996 


No. 585 L. 767.] 
prueioe | 2.) ab e*] sa oe wi wer 
| | | 
85 | 767156 | 7230 | 7304 | 7379 | 7453 || 7527 | 7601 | 7675 | 7749 
6 | 7898 | 7972 | 8046 8120 | 8194 8268 8342 | 8416 | 8490 
1 8638 | 8712 8786 8860 | 8934 || 9008 | 9082 | 9156 | 9230 
8 | 9877 | 9451 | 9525 9599 | 9673 || 9746 | 9820 | 9894 | 9968 
9 | 770115 | 0189 0263 | 0336 | 0410 | 0484 0557 | 0631 0705 
90 0852 | 0926 | 0999 | 1073 | 1146 1220 1293 | 1367 | 1440 
1 1587 | 1661 | 1734 | 1808 | 1881 || 1955 | 2028-) 2102 | 2175 
2 2399 | 2395 | 2468 | 2542 | 2615 | 2688 | 2762 | 2835 | 2908 
3 8055 | 3128 | 320i | 3274 | 3348 || 3421 | 3494 | 3567 | 3640 
4! 8786 | 3860 | 3933 | 4006 | 4079 || 4152 | 4225 | 4298 | 4371 | 
5 | 4517 | 4590 | 4663 | 4736 | 4809 | 4882 | 4955 | 5028 | 5100 | 
6 | 5246 | 5819 | 5392 | 5465 | 5538 | 5610 | 5683 | 5756 | 5829 | 
7| 5974 | 6047 | 6120 | 6193 | 6265 | 6388 | 6411 | 6483 | 6556 | 
8 | 6701 | 6774 | 6846 | 6919 | 6992 || 7064 | 7187 | 7209 | 7282 
9| 7427 | 7499 | 7572 | 7644 | 7717 | 7789 | 7862 | 7934 | 8006 
00 | 8151 | 8224 | 8296 | 83868 | 8441 | 8513 | 8585 | 8658 | 8730 
1| 8874 | 8947 | 9019 | 9091 | 9163 | 9286 | 9308 | 9380 | 9452 | 
2! 9596 | 9669 | 9741 | 9813 | 9885 |) 9957 |- : 
. | 0029 | 0101 | 0173 | 
3 | 780317 | 0389 | 0461 | 0533 | 0605 || 0677 | 0749 | 0821 | 0893 
4 1037 | 1109 | 1181 | 1253 | 1324 || 1396 | 1468 | 1540 | 1612 
5 | 1755 | 1827 | 1899 | 1971 | 2042 || 2114 | 2186 | 2258 | 2329 
6 | 2473 | 2544 | 2616 | 2688 | 2759 || 2831 | 2902 | 2974 | 3046 
7 | 8189 | 8260 | 3332 | 3403 | 3475 || 3546 | 3618 | 3689 | 3761 
8 | 3904 | 8975 | 4046 | 4118 | 4189 |! 4261 | 4332 | 4408 | 4475 
9 {| 4617 | 4689 | 760 | 4831 | 4902 | 4974 | 5045 5116 | 5187 
10 | 5380 | 5401 | 5472 | 5543 | 5615 || 5686 | 5757 | 5828 | 5899 | 
1 6041 | 6112 | 6183 | 6254 | 6325 || 6396 | 6467 | 6538 | 6609 | 
2| 6751 | 6822 | 6893 | 6964 | 7035 || 7106 | 7177 | 7248 | 7319 | 
8 | 7460 | 7531 | 7602 | 7673 | 7744 |) 7815 | 7885 | 7956 | 8027 | 
4| 8168 | 8239 | 8810 | 8881 | 8451 || 8522 | 8598 | 8663 | 8734 | 
5 8875-| 8946) 9016 | 9087 | 9157 || 9228 | 9299 | 9369 | 9440 
6 | 9581 | 9651 | 9722 | 9792 | 9863 || 9933. | | | 
| | | | | 0004 | 0074 | 0144 
7 | 790285 | 0356 | 0426 | 0496 | 0567 || 0637.) 0707 | 0778 | 0848 | 
8 | 0988 | 1059 | 1129 | 1199 | 1269 || 1840 | 1410 | 1480 | 1550 
9 1691 | 1761 | 1831 | 1901 | 1971 || 2041 | 2111 | 2181 | 2252 | 
20 | 2392 | 2462 | 2582 | 2602 | 2672 | 2742 | 2812 | 2882 | 2952 | 
1| 3092 | 3162 | 3231 | 3801 | 3371 || 8441 | 3511 | 3581 | 3651 | 
2] 3790 | 3860 | 3930 | 4000 | 4070 || 4189 | 4209 | 4279 | 4349 
3] 4488°| 4558 | 4627 | 4697 | 4767 || 4886 4906 | 4976 | 5045 | 
4| 5185 | 5254 | 5824 | 5893 | 5463 || 5532 | 5602 | 5672 | 5741 | 
5 | 5880 | 5949 | 6019 | 6088 | 6158 || 6227 | 6297 | 6366 | 6436 | 
6 | 6574 | 6644 | 6713 | 6782 | 6852 || 6921 | 6990 | 7060 | 7129 | 
7 | 7268 | 7337 | 7406 | 7475 | 7545 || 7614 | 7683 | 7752 | 7821 
8 | 7960 | 8029 | 8098 | 8167 | 8236 | 8305 | 8374 | 8443 | 8513 | 
9 | | 8858 | 8927 | 9065 | 9134 9203 | 


9 


7823 
8564 
9303 


0042 


0778 


1514 
2248 
2981 


| 8713 


4444 
5173 
5902 
6629 
7354 
8079 
8802 
9524 


9245 
0965 
1684 
2401 
3117 
3832 
4546 
5259 
5970 
6680 
7390 
8098 
8804 
9510 


0215 
0918 


| 1620 


2322 
3022 


721 
4418 
5115 
5811 
6505 
7198 
7890 
8582 


9272 


| 
| 
| 
| 


73 


oI 


70 


PROPORTIONAL PARTS. 


| DOMwOm ET 


| ROWROWEX 


ee BR OTOTort or 
DODO 
WOH OL 


[No. 629 L. 799. | 


TABE 


EK 


| No. 630 L. 799.] 


XXIV.—LOGARITHMS OF 


NUMBERS. 


[No. 674 L. 829. 


[N-| 0 | a" as 5 fae fa | ep. sal pim 
630 | 799341 | 9409 | 9478 | 9547 | 9616 || 9685 | 9754 | 9823 | 9892 | 9961 | 
1 | 800029 | 0098 | 0167 | 0236 | 0305 || 0373 | 0442 | 0511 | 0580 | 0648 | 
2 0717 | 0786 | 0854 | 0923 | 0992 |) 1061 | 1129 | 1198 | 1266 | 1335 | 
3 | 1404 | 1472 | 1541 | 1609 | 1678 ||'1747 | 1815 | 1884 | 1952 | 2021 | 
ee fate | Y 1 Ole OC | JON | 2021 | 
| 2089 pe | att | oe nite | 2432 | 2500 2568 | 2637 | 2705 
G | Bis7 | 35e5 | asgd | S002 | 3730 |) avas.| ssor | 3085 | 4008 | sock | 
; | 4139 | 4208 | 1976 | 4344 41412 || ri rit 3935 | 4003 | 4071 | 
| Oc Ad wid | € rs) } ¢ C | 5 8 ) > | 4685 | V5E i 
§ | 4821 | 4880 | 4057 | 5025 | 5093 |) 5161 | 229 | 5207 | 5365 | 54133 | 68 
9 | 5501 | 5569 | 5637 | 5705 | 5773 || 5841 | 5908 | 5976 | 6044 | 6112 | 
640 | 806180 | 6248 | 6316 | 6384 | 6451 || 6519-| 6587 | 6655 | 6723 | 6790 | 
OP] 1 Gia | 6025 OUD: | TORE, | TAG: || AE) rank | RL | TAO | 7a | 
3 | Gort | S270 | B10 | Sd | SiBt || S540 | 8616 | Sost | Brot | sHi8 | 
Ke Owtle oO xe | : Ax 2 
4 | 8886 | 8953 | 9021 | 9088 | 9156 || 9223 | 9290 0358 495 592 
5 | 9560 | 9627 | 9694 | 9762 | 9829 || 9896 | 9964 eal 
-——| - -——— | —| 0031 | 0098 | 0165 
6 | 810233 | 0300 0367 | 0434 | 0501 || 0569 | 0636 | 0703 | 0770 | 0837 
7 | 0904 | 0971 | 1039 | 1106 | 1173 || 1240 | 1307 | 1374 | 1441 | 1508 | @7 
8 | 1575 | 1642 | 1709 | 1776 | 1843 |) 1910 1977 | 2044 | 2111 | 2178 
9} 2245 | 2312 | 279 | 245 | 212 |) V9 | 2646 | 2713 | 2780 | 247 
| | 650 | 2918 | 2980 | 3047 | 3114 | 3181 || go47 | 3314 | 3881 | 3448 | 2514 
| a) eee) aoe | aber | daar | dei || aoe | aoee | aria | 4700 | 4817. 
| 2 248 | 4314 | 4381 7 | 4514 || 4581 | 4647 | 4714 | 4780 | 4847 
| 3 4913 | 4980 | 5046 | 5113 | 5179 || 5246 | 5312 | 5378 | 5445 | 5511 
| $ Poe pi ae aR oa || 5910 596 6042 | 6109 | 6175 
| 32 DBU0 é | : ID | 657% 3638 705 | 6771-1 68 
6 |. 6904 | 6970 | 7036 | 7102 | 7169 || ot 7301 fae? | 7433 | 7409 
| Bye nae 0070: | FONG: 1s 10a | CGR Pee ra ORE alls 
| : | = ee ree ie — : 2: ee ae | 8094 | 8160 
8 | 8226 | 8292 | 8358.| 8424 | 8490 || 8556 | 8622 | 8688 | 8754 | 8820 | 
| 9 | 8885 | 8951 | 9017 | 9083 | 9149 || 9215 | 9281 | 9346 o4t2 | 9478. |. % 
| | 660 | 9544 | 9610 | 9676 | 9741 | 9807 || 9873 | 9939 gar ti 
¥ | _ —|———| 0004 | 0070 | 0136 
| 1 | 820201 | 0267 0383 | 0399 | 0464 || 0530 | 0595 | 0661 | 0727 | o702 
2| 0858 | 0924 | 0989 | 1055 | 1120 || 1186 | 125 317 | 138% 18 
a] asd | 1570| 1048 vn0-| 1775 || 1841. | 1906 jo73-| 2087 | 2108 
4 | 2168 | 2233 | 2299 | 2364 | 2430 || 2495 one La DO ehee 
| £ | SES5 | Sat | dose | Sore | B00 || 3148 | aera | der9 | 33d | S400 
6 3474 | 3539 ae be Z JK BU69 3 Lat ¢ 213 8279 3344 | 3409 
| 6 3474 | 3538 3605 3670 | 3735 || 3800 | 3865 | 3930 | 3996 | 4061 
I | Z| 4126 | 4191 | 4256 | 4821 | 4386 |) 4451 | 4516 | 4581 4646 | 4711 | 
| 8 | 4776 | 4841 | 4906 | 4971 | 5036 |) 5101 | 5166 | 5231 5296 | 5361 | © 
9 | 5426 5491 | 5556 | 5621 | 5686 |) 5751 | 5815 | 5880 | 5945 | 6010 
| 670 | 6075 | 6140 | 6204 | 6269 | 6334 || 6399 | 6464 | 6528 | 6593 | 6658 
1| 6723 | 6787 | 6852 | 6917 | G9BL |) 7046 | 7111 | 7175 | 7240 | 7305 
2| 7369 | 7434 | 7499 | 7563 | 7628 || 7692 | 7757 | 7821 | 7886 | 7951 
3B} S015 | 8080 | 8144 | 8209 | 8273 | 8338 | 8402 | 8467 | 8581-) 8595 
4 | 8660 | 8724 | 8789 | 8853 | 8918 |) 8982 | 9046 | 9111 | 9175 | 9239 
| PROPORTIONAL PARTS. 
| Diff. | 1 2 a am ee 5 6 q 8 9 
se = 5 ; saya as 
| 68 | 6.8} 18.6 | 20.4 | 27.2 34.0 | 40.1 47.6 | 54.4 | 61.2 
/ 6% O.4 13.4 20.1 | 26.8 33.5 40.2 46.9 58.6 60.3 
| | 66 | 6.6 13.2 | 19.8 | 26.4 | 33.0 30.6 | 46.2 | 52.8 | 59.4 
| 6.5 3. 9.5} 26.0 32.5 389.0 Vo 52 58.5 
| 64 | 64] 12.8 | 19.2 | 25.6 32.0 | 38.4 4.8) 51.2 | 57.6 | 


© 


cr 


fom) 


TABLE XXIV.—-LOGARITHMS OF NUMBERS. 


No. 675 L. 829.] [No. 719 L. 857. 
pe 
woe Pop ita fee al ot ere ig AP ome 
. | bree | H | | 
| | | 
675 | 829304 | 9368 | 9432 | 9497 | 9561 || 9625 | 9690 | 9754 | 9818 | 9882 
6 9947 | = | Wecricdineel am icad Stee Saree 
————| 0011 | 0075 | 0189 | 0204 |} 0268 | 0382 | 0896 | 0460 | 0525 | 
@ | 8380589 | 0653 | 0717 | 0781 | 0845 |) 0909 | 0973 | 1087 | 1102 | 1166 | 
8 1230 | 1294 | 1358 | 1422 | 1486 || 1550 | 1614 | 1678 | 1742 | 1806 | 64 
9 1870 | 1984.) 1998 | 2062 | 2126 || 2189 | 2253 | 2817 | 2381 | 2445 | 
380 2509 | 2573") 2637 | 2700 | 2764 || 2828 | 2892 | 2956 |. 3020 | 3083 | 
1 8147 | 3211 | 3275 | 3338 | 3402 || 3466 | 3530 | 3593 | 3657 | 3721 | 
2 8/84 | 3848 | 3912 | 3975 | 4039 |; 4103 | 41€6 | 4230 | 4294 | 4357 | 
2 4421 | 4484 | 4548 | 4611 | 4675 |) 4739 | 4802 | 4866 | 4929 | 4993 | 
4 5056 | 5120 | 5183 | 5247 | 5310 |) 5873 | 5487 | 5500 | 5564 | 5627 | 
5 5691 | 5754 | 5817 | 5881 | 5944 || 6007 | 6071 | 6134 | 6197 | 6261 | 
6 6324 | 6887 | 6451 | 6514 | 6577 || 6641 | 6704 | 6767 | 6830 | 6894 | 
ie 6957 | 7020 | 7083 | 7146 | 7210 |) 7273 | 7336 | 7399 | 7462 | 7525 | 
8 88 | 7652 | 7715 | 7778 | 7841 || 7904 | 7967 | 8030 | 8093 | 8156 | 
9] 8219 | 8282 | 83845 | 8408 | 8471 || 8534 | 8597 | 8660 | 8723 | 8786 | 68 
690 8849 | 8912 | 8975 | 9038 | 9101 || 9164 | 9227 | 9289 | 9352 | 9415 
1 | 9478 | 9541 | 9604 | 9667 | 9729 || 9792 | 9855 | 9918 ) 9981 beer 
| | | | . | 0043 
2 | 840106 | 0169 | 0232 | 0294 | 0357 || 0420 | 0482 | 0545 | O608 | 0671 | 
3 0733 | 0796 | 0859 | 0921 | 0984 || 1046 | 1109 | 1172 | 1284 | 1297 
4} 1359 | 1422 | 1485 | 1547 | 1610 || 1672 | 1735 | 1797 | 1860 | 1922 
5 | 1985 | 2047 | 2110 | 2172 | 2235 || 2297 | 2360 | 2422 | 2484 | 2547 
6 | 2609 | 2672 | 2734 | 2796 | 2859 || 2921 | 2983 | 3046 | 3108 | 3170 
7 | 82383 | 8295 | 3357 | 3420 | 3482 || 3544 | 3606 | 3669 | 3731 | 3793 
8 | 8855 | 3918 | 3980 | 4042 | 4104 |) 4166 | 4229 | 4291 | 4353 | 4415 
9 | 77 | 4539 | 4601 | 4664 | 4726 || 4788 | 4850 | 4912 | 4974 | 5036 
700 | 5098 | 5160 | 5222 | 5284 | 5346 || 5408 | 5470. | 5532 | 5594 | 5656 62 
1 5718 | 5780 | 5842 | 5904 | 5966 || 6028 | 6090 | 6151 | 6213 | 6275 
2 | 6837 | 6399 | 6461 | 6523 | 6585 | 6646 | 6708 | 6770 | 6832 | 6894 
3 | - 6955 | 7017 | 7079 | 7141 | 7202 || 7264 | 7326 | 7388 | 7449 | 7511 
4 1573 | 7634 | 7696 | 7758 | 7819 | 7881 | 7943 | 8004 | 8066 | 8128 
5 | 8189 | 8251 | 8312 | 8374 | 8435 | 8497 | 8559 | 8620 | 8682 | 8743 
6 | 8805 | 8866 | 8928 | 8989 | 9051 || 9112 | 9174 | 9285-| 9297.) 9858 
7 | 9419 | 9481 | 9542 | 9604 | 9665 | 9726 | 9788 | 9849 | 9911 | 9972 
8 | 850033 | 0095 | 0456 | 0217 | 0279 || 0340 | 0401 | 0462 | 0524 | 0585 
9 | 0646 | 0707 | 0769 | 0880 | 0891 | 0952 | 1014 | 1075 | 1136 | 1197 
710 | 1258 | 1820 | 1881 | 1442 | 1503 || 1564 | 1625 | 1686 | 1747 | 1809 
1 | 1870 | 1931 | 1992 | 2053 | 2114 || 2175 | 2286 | 2297 | 2358 | 2419 
2 | 2480 | 2541 | 2602 | 2663 | 2724 || 2785 | 2846 | 2907 | 2968 | 3029 61 
3; 8090 | 3150 | 8211 | 3272 | 3333 || 3394 | 3455 | 3516 | 3577 | 3637 
4 3698 | 3759 | 3820 | 3881 | 3941 || 4002 | 4063 | 4124 | 4185 | 4245 
5 | 4306 | 4367 | 4428 4488 | 4549 | 4610 | 4670 | 4731 | 4792 | 4852 
6 | 4913 | 4974 | 5034 | 5095 | 5156 || 5216 | 5977 | 5337 | 5398 | 5459 | 
7 5519 | 5580 | 5640 | 5701 | 5761 | 5822 | 5882 | 5943 | 6003 | 6064 
8 6124 | 6185 | 6245 | 6806 | 6366 | 6427 | 6487 | 6548 | 6608 | 6668 
9 | 6729 | 6789 | 6850 | 6910 | 6970 || 7031 | 7091 | 7152 | 7212 | 7272 
| | | | 
PROPORTIONAL PARTS. 
- Nl j 
Diff. | 1 2 3 4 | 5) 6 q 8 9 
65 | 6.5 | 13.0 19.5 | 26.0 | 32.5 39.0 45.5 52.0 58.5 
64. | 6.4] 12.8 19.2 25.6 | 32.0 38.4 44.8 51.2 57.6 | 
63 6.3 | 12.6 | 18.9 25.2 | 81.5 | 387.8 44.1 50.4 56.7 
62 6.2 | 12.4 18.6 24.8 | 31.0 | 37.2 | 43.4 49.6 55.8 
61 | 6.1] 12.2 18.3 24.4 | 30.5 36.6 2.7 48.8 54.9 
60 6.0} 12.0 | 18.0 | 24.0 | 30.0 36.0 | 42.0 48.0 | 54.0 


a a ee ee ee 


ee 


| 
| 


Tee 


———<—— 


L 


2 |2| 


TABLE XXIV.—LOGARITHMS OF 


NUMBERS. 


[No. 764 L. 883. 


No, 720 L. 857.] 


6 


9 | Diff. 


8297 


9499 


7694 

8417 | 8477 
9018 | 9078 
9619 | $679 60 


8898 


5 | (875 


0098 
0697 
1295 
1893 
2489 
3085 
3680 
4274 
4867 
5459 
6051 
6642 
7282 
7821 
8409 
8997 
9584 


0170 

15D 
1339 
1923 
2506 
3088 
3669 
4250 
4830 
5409 
5987 
6564 
7141 


0218 | 0278 
0817 | 0877 
1415 | 4475 
2012 | 2072 
2608 | 2668 
3204 | 3263 
99 | 3858 
4392 | 4452 
4985 | 5045 
5578. | 5687 
6169 | 6228 
6760 | 6819 
350 | 7409 | 99 
939 | 7998 
8527 | 8586 | 
9114 | 9113 


01 ; 9760 


287 | 0345 
72 | 09380 
1456 | 1515 
2040 | 2008 
2622 | 2681 
8204 | 38262 
85 | 3844 


| 5524 | 5582 
6102 | 6160 
6680 | 6737 
D6 | 7314 
832 | 7889 
8407 | 8464 
8981 | 9039 
9555 | 9612 


0127 | 0185° 


4366 | 4424 58 
4945 | 5003 


0699 | 0756 
1271 | 1328 
1841 | 1898 2 
2411 | 2468 | 5% 
2980 | 3037 
3548 | 3605 


0 1 | 2 

857332 | 7393 | 7453 
7935 | 7995 | 8056 
8537 | 8597 | 8557 
9138 | 9198 | 9258 

9739 | 9799 | 9859 | 
5 | 860338 | 0398 | 0458 
6 0937 | 0996 | 1056 
i 1534 | 1594 | 1654 
8 2131 | 2191 | 2251 
9 |. 2728 | 2787 | 2847 
730 | 3323 | 8382 | 3442 
1} 8917 | 38977 | 4036 
21 4511 | 4570 | 4630 
3 5104 | 5163 | 5222 
4| 5696 | 5755 | 5814 
5 6287 | 6346 | 6405 
6 | 6878 | 69387 | 6996 
%| 7467 | 7526 | 7585 
8 | 8056 | 8115 | 8174 
9 | 8644 | 8703 | 8762 
740 | 9232 | 9290 | 9349 
1| 9818 | 9877 | 9935 
2 | 870404 | 0462 | 0521 
3 0989 | 1047 | 1106 
4 1573 | 1631. | 1690 
5 2156 | 2215. | 2273 
6 9739 | 2797 | 2855 
va 3321 | 3379 | 3437 
8 3902 | 3960 | 4018 
9 | 4482 4540 | 4598 
750 5061 | 5119 | 5177 
1 | 5640 | 5698 | 5756 
Q 6218 | 6276 | 6333 
3 6795 | 6853 | 6910 
4 7371 | 7429 | 7487 
5 7947 | 8004 | 8062 
6 8522 | 857 8637 
7% | 9096 | 9153 | 9211 
8 | 9669 | 9726 | 9784 
9 | 880242 | 0299 | 0356 
760 0814 | 0871 | 0928 
1 | 1385 | 1442 | 1499 
2 1955 | 2012 | 2069 
B} 9525 | 2581 | 2638 
4 3093 | 3150 | 3207 

1 2 


GO G0 6 
Sao | 


0  W%* 


8 9 
47.2 53.1 
46.4 52.2 
45 .6 51.3 
44.8 50.4 


$$$ 


TABLE XXIV.—LOGARITHMS OF NUMBERS. 


No. 765 L. 883.] [No. 809 L. 908. 


N.| 0 i;/2)}% /4 6 | ® | & | 8) 9) | pif. 
765 | 883661 | 3718 | 3775 | 3832 | 3888 | 38945 | 4002 | 4059 | 4115 | 4172 | | 
6 422 4285 | 4842 | 4899 | 4455 4512 | 4569 | 4625 | 4682 | 4739 | | 
ton 4°95 | 4852 | 4909 | 4965 | 5022 || 5078 | 5135 | 5192 | 5248 | 5305 
8 5861 | 5418 | 5474 | 5531 || 5587 5644 | 5700 | 5757 | 5813 | 5870 
9 5926 | 5983 | 6039 | 6096 | 6152 || 6209 | 6265-| 6321 | 6878 | 6434 | 
770 6491 | 6547 | 6604 | 6660 | 6716 || 6773 | 6829 | 6885 | 6942 | 6998 
hea} 7054 | 7111 | 7167 | 7223 | 7280 73386 | 73892 | 7449 | 7505 | 7561 
2 7617 | 7674 | 7730 | 7786 | 7842 7898 | 7955 | 8011 | 8067 | 8123 
3 8179 | 8236 | 8292 | 8348 | 8404 8460 | 8516 | 857% 8629 | 8685 
4 8741 | 8797 | 8853 | 8909 | 8965 9021 | 9077 |. 9184 | 9190 | 9246 
5 9302 | 9358 | 9414 | 9470 | 9526 || 9582 | 9688 | 9694 | 9750 | 9806 56 
6 9862 | 9918 | 9974 . —— — 
0030 | 0086 0141 | 0197 | 0253 | 0309 | 0365 
7 | 890421 | 0477 | 05383 | 0589 | 0645 0700 | 0756 | 0812 | 0868 | 0924 
ih 8 0980 | 1035 | 1091 | 1147 | 1203 1259 | 1314 |-13870 | 1426 | 1482 
9 1537 | 1593 | 1649 | 1705 | 1760 1816 | 1872 | 1928 | 1983 | 2039 
fa 780 2095 | 2150 | 2206 | 2262 | 2317 23873 | 2429 | 2484 | 2540 | 2595 
i i 2651 | 2707 | 2762 | 2818 | 2873 || 2929 | 2985 | 8040 | 8096 | 3151 
nt | 2 8207 | 3262 | 3318 | 3373 | 3429 || 3484 | 3540 | 3595 | 3651 706 
(rea 3 3762 | 3817 | 38873 | 38928 |. 3984 || 4089 | 4094 | 4150 205 | 4261 
Hy ie || 4 4316 | 4871 | 4427 | 4482 | 4538 4593 | 4648 | 4704 | 4759 | 4814 
Cal 5 4870 | 4925 | 4980 | 5036 | 5091 5146 | 5201 | 5257 | 53812 | 5367 
u 6 5423 | 5478 | 5533 | 5588 | 5644 5699 | 5754 | 5809 | 5864 | 5920 
{ | ff 5975 | 6030 | 6085 | 6140 | 6195 6251 | 6806 | 63861 | 6416 | 6471 
i 8 6526 | 6581 | 6686 | 6692 | 6747 6802 | 6857 | 6912 | 6967 | 7022 
i| 9 7077 | 7182 | 7187 | 7242 | 7297 || 73852 | 7407 | 7462 | 7517 | 7572 55 
Hat 790 7627 | 7682 | 7737 | 7792 | 7847 7902 | 7957 | 8012 | 8067 | 8122 
ips | 1 8176 | 8231 | 8286 | 8341 | 8396 8451 | 8506 | 8561 | 8615 | 8670 
Hh 2 3725 | 8780 | 8835 | 8890 | 8944 8999 | 9054 | 9109 | 9164 | 9218 
et | 3 9273 | 9828 | 9383 | 9487 |. 9492 9547 | 9602 | 9656, | 9711 | 9766 
Rint | 4 9821 | 9875 | 99380 | 9985 —_—— | —_ =e _——|——_—— 
at — 00389 0094 | 0149 | 0208 | 0258 | 0312 
| 5 | 900867 | 0422 | 0476 | 0531 | 0586 || 0640 | 0695 749 | 0804 | 0859 
i i \\ 6 0913 | 0968 | 1022 | 1077 | 1181 1186 | 1240 | 1295 | 1349 | 1404 
i i t 1458 | 1518 | 1567 | 1622 | 1676 1731 | 1785 | 1840 | 1894 | 1948 
8 2003 | 2057 | 2112 | 2166 | 2221 2275 | 2829 | 2384 | 2488 | 2492 
H, | 9 2547 | 2601 | 2655 | 2710 | 2764 2818 | 2873 | 2927 | 2981 | 30386, 
id i 800 8090 | 3144 | 3199 | 8258 | 3307 38361 | 3416 | 3470 | 8524 | 3578 
tt Ia | 3633 | 3687 | 38741 | 3795 | 3849 8904 | 3958 | 4012 | 4066 | 4120 
if i 2 4174 | 4229 | 4283 | 4337 | 4891 4445 | 4499 | 4558 | 4607 | 4661 
| 3 4716 | 477 4824 | 487 4932 4986 | 5040 | 5094 | 5148 | 5202 54 
al 4 5256 | 5810 | 5864 | 5418 | 5472 5526 | 5580 | 5634 | 5688 | 5742 
Ut a 5 796 | 5850 | 5904 | 5958 | 6012 6066 | 6119 | 617% 6227 | 6281 
ana | 6 63835 | 6389 | 6443 | 6497 | 6551 6604 | 6658 | 6712 | 6766 | 6820 
| 74 6874 | 6927 | 6981 | 7035 | 7089 || 7143 | 7196 | 7250 | 73804 | 7358 
Ai 8 7411 | 7465 | 7519 | 7578 | 7626 ||: 7680 | 7734 | 7787 | 7841 | 7895 
9 7949 | 8002 | 8056 | 8110 | 8163 || 8217 | 8270 | 8324 | 837, 8431 
i | 
i 
PROPORTIONAL PARTS. | 
| | 
Diff 1 2 3 4 5 6 ve 8 9 | 
2 BN fe Sat — ies kittie ah Sa, Shee aan 
I} BY 5.7 11.4 Ved 22.8 28.5 34.2 39.9 45.6 51.3 
| 56 os0 ee 16.8 22.4 28.0 33.6 39.2 44.8 50.4 | 
5.5 . 16.5 22.0 27.5 44.0 49.5 
5.4 16.2 21.6 7.0 43 .2 48.6 


TABLE XXTV.—lIOGARITHMS 


OF 


NUMBERS. 


[No. 854 L. 981. | 


No. 810 L. 908.] 

N.| 0 1 2 3 4 | Bil 8 q 8 
810 | 908485 | 8539 | 8592 | 8646 | 8699 || 8753 | 8807 | 8860 | 8O14 
ai 9021 | 9074 | 9128 | 9181 | 92385 {| 9289 | 9342 | 9396 | 9449 

2} 9556 | 9610 | 9663 | 9716 | 9770 || 9823 | 9877 | 9930 | 9984 
3 | 910091: | 0144 | 0197 | 0251 | 0304 |) 0358 | 0411 | 0464 | 0518 
4 | 0624 | 0678 | 0731 | 0784 | 0838 || 0891 | 0944 | 0998 | 1051 
5 | 1158] 1211 | 1264 | 1317 | 1371 || 1424 | 1477 | 1530 | 1584 
6 | 1690 | 1743 | 1797 | 1850 | 1903 || 1956 | 2009 | 2063 | 2116 
7 | 2aR2 | 2275 | 2328 | 2381 | 2435 || 2488 | 2541 | 2594 | 2647 
8 | 2753 | 2806 | 2859 | 2913 | 2966 || 3U19 | 8072 | 8125 | 3178 
9 | 3284 | 3337 | 3390 | 3443 | 8496 || 3549 | 8602 | 3655 | 3708 
g20 | 3814 | 3867 | 3920 | 3973 | 4026 || 4079 | 4132 | 4184 | 4287 
1| 4343 | 4396 | 4449 | 4502 | 4555 |! 4608 | 4660 | 4713 | 4766 
2| 4872 | 4925 | 4977 | 5030 | 5083 || 5136 | 5189 | 5241 | 5294 
3.| 5400 | 5458 | 5505 | 5558 | 5611 || 5664 | 5710 | 5769 | 5822 
4| 5927 | 5980 |.6033 | 6085 | 6138 || 6191 | 6243 | 6296 | 6349 
5 | 6454 | 6507 | 6559 | 6612 | 6664 || 6717 | 6770 | C822 | 6875 
6 | 6980 | 7033 | 7085 | 7138 | 7190 || 7243 | 7295 | 7348 | 7400 
7 | 7506 | 7558 | 7611-| 7663 | 7716 || 7768 | 7820 | 7873 | 7925 
8 | 8030 | 8083 | 8135 | 8188 | 8240 || 8293 | 8345 | 8897 | 8450 
9 | 8555 | 8607 | 8659 | 8712 | 8764 || 8816 | 8869 | S921 | 8973 
830 | 9078 | 9130 | 9188 | 9235 | 9287 || 9340 | 9392 | 9444 | 9496 
1| 9601 | 9653 | 9706 | 9758 | 9810 || 9862 | 9914 | 9967 os 
ACRE | pert elgg Pe cag ph) Sa 0015 

2 | 920123 | 0176 | 0228 | 0280 | 0382 || 0384 | 0436 | 0489 | 0541 
3| 0645 | 0697 | 0749 | 0801 | 0853 || 0906 | 0958 | 1010 | 1062 
4| 1166 | 1218 | 1270 | 1322 | 1374 || 1426 | 1478 | 1580 | 1582 
| 5 | 1686 | 1738 | 1790 | 1842 | 1894 |} 1946 | 1998 | 2050 | 2102 
| 6 | 2206 | 2258 | 2310 | 2362 | 2414 || 2466 | 2518 | 2570 | 2622 
| v | 9795 | 9777 | 2829 | 2881 | 2933 || 2985 | 3087 | 3089 | 3140 
g | 3244 | 3296 | 3348 | 3399 | 3451 || 3503 | 3555 | 3607 | 3658 
9 | 38762 | 8814 | 8865 | 8917 | 3969 || 4021 | 4072 | 4124 | 4176 
840 | 4279 | 4331 | 4383 | 4434 | 44s6 || 4538 | 4589 | 4641 | 4693 
1| 4796 | 4848 | 4899 | 4951 | 5008 || 5054 | 5106 | 5157 | 5209 
21 5312 | 5364 | 5415 | 5467 | 5518 || 5570 | 5621 | 5673 | 5725 
3 | 5828 | 5879 | 5931 | 5982 | 6034 || 6085 | 6137 | 6188 | 6240 
4 | 6342 | 6394 | 6445 | 6497 | 6548 || 6600 | 6651 | 6702 | 6754 
5 | 6857 | 6908 | 6959 | 7o11 | 7062 || 7114 | 7165 | 7216 | 7268 
6 | 7370 | 7422 | 7473 | 7524 | w576 |) 7627 | Te78 | 7730 | 7781 
7 |  7g93 | 7935 | 7986 | 8037 | 8088 || 8140 | 8191 | 8242 | 8293 
8 | 8396 | 8447 | 8498 | 8549 | 8601 || 8652 | 8703 | 8754 8805 
9 | 8908 | 8959 | 9010 | 9061 | 9112 || 9163 | 9215 | 9266 | 9317 
850 | 9419 | 9470 | 9521 | 9572 | 9623 || 9674 | 9725 | 9776 | 982% 
1 | 9930 | 9981 18 = Et | ie 
—__|____| 9032 | 0083 | 0134 || 0185 | 0236 | 0287 | 0338 
2 | 930440 | 0491 | 0542 | 0592 | 0643 || 0694 | 0745 | 0796 | 0847 
31 0949 | 1000 | 1051 | 1102 | 1153 || 1204 | 1254 | 1805 | 1356 
4| 1458 | 1509 | 1560 | 1610 | 1661 || 1712 | 1763 | 1814 | 1865 


Diff. 


53 


TABLE XXIV.—LOGARITHMS OF NUMBERS. 
| No. 855 L. 931.] [No, 899 L. 954, | 
N. | 0 2 5 6 7 8 Diff. 
855 | 931966 | 2017 | 2068 1 2220 | 2822 | 2372 | 
6 | 2474 | 2524 | 2575 | 2727 2829 | 2879 
7 | 2981 | 3031 | 3082 | 32 3335 | 3386 | 3437 
8 | 3487 | 3538 | 3589 | 3740 | 3841 | 3892 
9 3993 | 4044 | 4094 (4246 | 4347 | 4397 | 
860 | 4498 | 4549 | 4599 | 4051 | 4852 | 4902 
1 | 5003 | 5054 | 5104 || 5255 5356 | 5406 | 
2| 5507 | 5558 | 5608 || 5759 5860 | 5910 
3| 6011 | 6061 | 6111 6262 3363 | 6413 
4| 6514 | 6564 | 6614 6765 6865 | 6916 
5 | 7016 | 7066 | 7116 7267 | 7367 | 7418 | 
6 7518 | 7568 | 7618 7769 | 7869 | 7919 BO 
7 | 8019 | 8069 | 8119 | 8269 83870 | 8420 
8} 8520 | 8570 | 8620 877 | 8870 | 8920 
9} 9020 | 9070 | 9120 927 9869 | 9419 
870 | 9519 | 9569 | 9619 9769 | 9869 | 9918 
1 | 940018 | 0068 | 0118 | 0267 0367 | 0417 
2| 0516 | 0566-| 0616 | 0765 | 0865 | 0915 
3 1014 | 1064 | 1114 1263 | 1362 | 1412 
4| 1511 | 1561 | 1611 1760 | 1859 | 1909 
5 | 2008 | 2058 | 2107 | 2256 | 2355 | 2405 
6 | 2504 | 2554 | 2603 || 2752 2851 | 2901 | 
7 | 3000 | 3049 | 3099 3247 | 3346 | 3396 
8 | 3495 | 3544 | 3593 3742 3841 | 3890 
9 | 3989 | 4038 | 4088 | 4236 4335 | 4384 
880 | 4483 | 4582 | 4581 | 4729 | 4828 | 4877 
1| 4976 | 5025 | 5074 || 5222 5821 | 5370 
2| 5469 |.5518 | 5567 | 5715 5813 | 5862 
3| 5961 | 6010 | 6059 6207 | 68305 | 6354 
4| 6452 | 6501 | 6551 6698 | 6796 | 6845 
5 | 6943 | 6992 | 7041 7189 | 7287 | 7336 49 
6| 7434 | 7483 | 7582 | 7679 | 7777 | 7826 
7| 7924 | 7973 | 8022 8168 | | 8266 | 8315 
8} 8413 | 8462 | 8511 8657 | | 8755 | 8804 
9} 8902 | 8951 | 8999 9146 | 9244 | 9292 
; 890 | 9390 | 9439 | 9488 | 9634 | 9731 | 9780 
1]. 9878 | 9926 | 9975 | [aes | 
/ | || 0121 | 0219 | 0267 
2 | 950365 | 0414 | 0462 0608 0706 | 0754 
3} 0851 | 0900 | 0949 1095 | 1192 | 1240 
4} 1888 | 1386 | 1435 | 1580 1677 | 1726 
5 | 1823 | 1872 | 1920 | 2066 | 2163 | 2211 
G6 | .2308 | 2356 | 2405 2550 2647 | 2696 
7 | 2792 | 2841 | 2889 | 3034 | 3131 | 3180 
8 | 8276 | 3325 | 3373 3518 | 3615 | 3663 
9 | 3760 | 3808 | 3856 4001 | 4098 | 4146 


TABLE XXTIV.—LOGARITHMS OF NUMBERS. 


i] 
|No 900 1. 954.1] [No. 944 L. 975. 
| | 
IN. | 0 Pi Pie | ai ae | ep 3 8 | 9 Diff. 
| | 
| | | 
| 900 | 954243 | 4291 | 4339 | 4387 | 4435 || 4484 | 4582 | 4580 | 4628 | 4677 
rod 4725 | 4773 | 4821 | 4869 | 4918 || 4966 | 5014 | 5062 | 5110 | 5158 
We 2 5207 | 5255-| 5303 | 5351 | 5399 || 5447 | 5495 | 5543, 5592 | 5640 
| 31 5688 | 5736 | 5784 | 5882 | 5880 || 5928) 5976 | 6024 | 6072 , 6120 
| 4 | 6168 | 6216 | 6265 | 6313 | 6361 || 6409 | 6457 | 6505 | 6553 | 6601 48 
| 5 | 6649 | 6697 | 6745 | 6793 | 6840 :| 6888 | 6936 | 6984 | 7032 | 7080 
| 6! 7128 | 7176 | 7224 | 7272 | 7820 || 7368 | 7416 | 7464 | 7512 | 7559 
rf 7607 | 7655 | 7703 | 7751 | 7799 | 7847 7894 | 7942 | 7990 | 8038 
8! goxG | 8134 | 8181 | 8229 | 827% || 8825 | 83873 | 8421 | 8468 | 8516 
9 | 8564 | 8612 | 8659 | 8707 | 8755 || 8803 | 8850 | 8898 | 8946 | 8994 
910 | 9041 | 9089 | 9137 | 9185 | 9232 | 9280 | 9828 | 9375 | 9423 | 9471 
| 1] 9518 | 9566 | 9614 | 9661 | 9709 || 9757 | 9804 | 9852 | 9900 | 9947 
| 2} 9995 | = | — = 
| | 0042 | 0090 | 0188 | 0185 || 0283 | 0280 | 0328 | 0876 | 0423 
3 | 960471 | 0518 | 0566 | 0613 | 0661 || 0709 | 0756 | 0804 | 0851 | 0899 
4 | 0946 | 0994 | 1041 | 1089 | 1136 || 1184 | 1231 | 1279 1826 | 1374 
5 | 1421 | 1469 | 1516 | 1563 | 1611 || 1658 | 1706 | 1753 | 1801 | 1848 
| 61 1895 | 1943 | 1990 | 2088 | 2085 || 2182 | 2180 | 2227 | 2275 | 23822 
7 | 2369 | 2417 | 2464 | 2511 | 2559 || 2606 | 2653 | 2701 | 2748 | 2795 
8 | 9948 | 9890 | 2937 | 2985 | 3082 || 3079 | 3126 | 3174 | 3221 | 8268 
| 9 | 3316 | 3863 | 3410 | 3457 | 3504 |) 3552 | 3599 | 3646 | 3693 | 3741 
920 3788 | 3835 | 3882 | 3929 | 2977 || 4024-) 4071 | 4118 | 4165 | 4212 
1| 4260 | 4307 | 4354 | 4401 | 4448 || 4495 | 4542 | 4590 | 4637. | 4684 
2 4731 | 4778 | 4895 | 4872 | 4919 || 4966 | 5013 | 5061 . 5108 | 5155 
| 3 | 5202 | 5249 | 5296 | 5343 | 5390 || 5437 | 5484 | 5531 5578 | 5625 
| 4 5672 | 5719 | 5766 | 5813 | 5860 || 5907 | 5954 | 6001 | 6048 | 6095 4g 
5 | 6142 | 6189 | 6236 | 6283 | 6329 |) 6376 | 6423 | 6470 | 6517 | 6564 
6 | 6611 | 6658 | 6705 | 6752 | 6799 || 6845 | 6892 | 6939 | 6986 | 7033 
7 "ogo | 7127 | 71'73.| 7220 | 7267. || 7314 | 7361 | 7408 | 7454 | 7501 
8 rag | 7595 | 7642 1 7688 | 7735 || 7782. | 7829 | 7875 | 7922 | 7969 
a 8016 | 8062 | 8109 | 8156 | 8203 || 8249 | 8296 | 8343 | 8390 | 8436 
930 8483 | 8530 | 8576 | 8623 | 8670 || 8716 | 8763 | 8810 | 8856 | 8903 
1 | 3950 | 8996 | 9043 | 9090 | 9136 || 9183 | 9229 | 9276 | 9323 | 9369 
2} 9416 | 9463 | 9509 | 9556 | 9602 || 9649 | 9695 | 9742 | 9789 | 9885 
3 | 9882 | 9928 | 9975 | La 
| | | 0021 | 0068 |} O114 | 0161 | 0207 | 0254 | 0300 
4 | 970347 | 0393 | 0440 | 0486 | 0533 || 0579 | 0626 | 0672 | 0719 | 0765 
| 5 0812 | O858 | 0904 | 0951 | 0997 || 1044 | 1090 | 1137 | 1183 | 1229 
6| 1276 | 1322 | 1869 | 1415 | 1461 || 1508 | 1554 | 1601 | 1647 | 1693 
| 7 | 1740 | 1786 | 1882 | 1879 | 1925 || 1971 | 2018 | 2064 | 2110 | 2157 
8 2903 | 2249 | 2295 | 2342 | 2388 || 2434 | 2481 | 2527 | 2573 | 2619 
9 | 2666 | 2712 | 2758 | 2804 | 2851 |} 2897 | 2943 | 2989 | 8035 | 3082 
| | | | 
1940 | 3128 | 3174 | 3220 | 3266 | 3313 || 3359 | 3405 | 3451 | 3497 | 3543 
iP ot 3590 | 3636 | 3682 | 3728 | 3774 || 3820 | 3866 | 3913 | 3959 | 4005 
| 2 4051 | 409% | 4143 | 4189 | 4235 || 4281 | 4827 | 4874 | 4420 | 4466 
3! 4512 | 4558 | 4604 | 4650 | 4696 || 4742 | 4788 | 4834 | 4880 | 4926 
| 4} 4972 | 5018 | 5064 | 5110 | 5156 |} 5202 | 5248 | 5294 | 5340 | 5386 46 | 
| if 
PROPORTIONAL PARTS. 
| | | poe 
| Diff | 1 | 2 | 3 sat 15.5 6 Fe 8 9 
} | 
el aa 2a als ead wed wk | 
| AG | ANG OM | 14.1 18.8 | 23.5 28 2 32.9 37.6 42.3 | 
| 4 | 40] 92 | 18 | 184 | 23.0 | a6 | a2 | 36.8 41.4 | 


{ 


so 
or 


CRS SONI PWWHS S® 


OOS Ot Co 


BS 
S 


COOIMopomwHS CO VNIOounrwmore 


TABLE XXTIV.—LOGARITHMS OF NUMBERS. 
a ee 
No. 945 L. 975.] 


0 1 2 3 4 5 6 v4 
975432 | 5478 | 5524 | 5570 | 5616 || 5662 | 5707 | 5753 
5891 | 6937 | 5983 | 6029 | 6075 || 6121 | 6167 | 6212 
6350 | 6896 | 6442 | 6488 | 6523 || 6579 | 6625 | 6671 
6808 | 6854 | 6900 | 6946 | 6992 || 7037 | 7088 | 7129 
7266 | 7312 | 73858 | 7403 | 7449 || 7495 | 7541 | 7586 
TRA | 7769 | 7815 | 7861 | 7906 || 7952 | 7998 | 8043 
8181 | 8226 | 8272 | 8317 | 8363 || 8409 | 8454 | 8500 
8637 | 8683 | 8728 | 8774 | 8819 || 8865 | 8911 | 8956 
9093 | 9188 | 9184 | 9230 | 9275 || 9321 | 9366 | 9412 
9548 | 9594 | 9639 | 9685 | 9730 || 9776 | 9821 | 9867 
980003 | 0049 | 0094 | 0140 | 0185 || 0231 | 0276 | 0322 
(458 | 0503 | 0549 | 0594 | 0640 || 0685 | 0730 | 0776 
0912 | 0957 | 1003 | 1048 | 1093 || 1189 | 1184 | 1229 
1366 | 1411 | 1456 | 1501 | 1547 || 1592 | 1637 | 1683 
1819 | 1864 | 1909 | 1954 | 2000 '| 2045 | 2090 | 2135 
2271 | 2316 | 23862 | 2407 | 2452 || 2497 | 2543 | 2588 
2723 | 2769 | 2814 | 2859 | 2904 || 2949 | 2994 | 3040 
3175 | 3220 | 8265 | 3310 | 3856 || 3401 | 3446 | 3491 
3626 | 3671 | 3716 | 3762 | 8807 || 3852 | 3897 | 3942 
4077 | 4122 | 4167 | 4212 | 4257 |) 4802 | 4347 | 4392 
4527 | 4572 | 4617 | 4662 | 4707 752 | 4797 | 4842 
4977 | 5022 | 5067 | 5112 | 5157 || 5202 | 5247 | 5292 
5426 | 5471 | 5516 | 5561 | 5606 || 5651 | 5696 | 5741 
5875 | 5920 | 5965 | 6010 | 6055 || 6100 | 6144 | 6189 
6324 | 6369 | 6413 | 6458 | 6503 || 6548 | 6593 | 6637 
6772 | 6817 | 6861 | 6906 | 6951 || 6996 | 7040 | 7085 
7219 | 7264 | 7209 | 7353 | 7398 || 7443 | 7488 | 7582 
7666 | 7711 | 7756 | 7800 | 7845 || 7890 | 7934 | 7979 
8113 | 8157 | 8202 | 8247 | 8291 || 8336 | 8381 | 8425 
8559 | 8604 | 8648 | 8693 | 8737 || 8782 | 8826 | 8871 
9005 | 9049 | 9094 | 9138 | 9183 || 9227 | 9272 | 9316 
9450 | 9494 | 9539 | 9583 | 9628 || 9672 | 9717 | 9761 
9895 | 9939 | 9983 _ 
0028 | OO72 |} 0117 | 0161 | 0206 
990329 | 0383 | 0428 | 0472 | 0516 || 0561 | 0605 | 0650 
0783 | 0827 | 0871 | 0916 | 0960 || 1004 | 1049 | 1098 
1226 | 1270 | 1315 | 13859 | 1403 || 1448 | 1492 | 1536 
1669 | 1713 | 1758 | 1802 | 1846 || 1890 | 1935 | 1979 


[No. 989 L. 995. | 


Diff. 


45 


~J 
(=r) 
22 
ris) 


| No. 990 L. 995.] 


TABLE XXTV.—LOGARITHMS OF NUMBERS. 


[No. 999 LL. 999. 


Nei 8 1 2 3 4 5 6 3 8 9 | Diff. 
| eS —_——_—- | oo - —————- ———— 
990 | 995635 | 5679 W283 | 5767 | 5811 |, 5854 | 5898 | 5942 | 5986 | 6030 
1 6074 | 6117 | 6161 | 6205 | 6249 |. 6293 | 6337 | 6880 | 6424 | 6468 44 
2| 6512 | 6555 | 6599 | 6643 | 6687 | 6731 | 6774 | 6818 | 6862 | 6906 
3 | 6949 | 6993 | 7037 | 7080 | 7124 | 7168 | 7212 | 7255 | 7299 | 7343 
4 | 386 | 7430 | 7474 | V51T | 7561 | 7605 | 7648 | (692 | 7736 | TTT 
5 | 7823 | 7867 | 7910 | 7954 | 7998. 8041 | 8085 | 8129 | 8172 | 8216 
6 | 8259 | 8803 | 83847 | 8390 | 8434 | 8477 | 8521 | 8564 | 8608 | 8652 
7 | 8695 | 8739 | 8782 | 8826 | 8869 | 8913 | 8956 | 9000 | 9048 9087 
8 | 9131 | 9174 | 9218 | 9261 | 9305 | 9848 | 9392 | 9485 | 9479 | 9522 
9 | 9565 | 9609 | 9652 | 9696 | 9739 | 9783 | 9826 | 9870 | 9913 | 9957 | 4g 
| | | | 
LoGARITHMS OF NUMBERS FROM 1 To 100. 
\| | | 
N. Log. || N.| Log. Wot Logo N.. ik Log. <1 Nye Log 
1 | 0.000000 || 21 | 1.322219 || 41 | 1.612784 || 61 | 1.785330 || 81 | 1.908485 
2 | 0.301030 || 22 | 1.342428 || 42 | 1.623249 || 62 | 1.792392 || 82 | 1.918814 
3| 0.477121 || 23 | 1.861728 || 43 | 1.633468 || 63 | 1.799341 || 83 | 1.919078 
4 0.602060 || 24 | 1.880211 || 44 | 1.643453 || 64 1.806180 |) 84 | 1.924279 
5 | 0.698970 || 25 | 1.897940 |} 45 | 1.658218 || 65 1.812918 || 85 | 1.929419 
6 | 0.778151 || 26 | 1.414973 || 46 | 1.662758 || 66 | 1.819544 || 86 | 1.934498 
7 | 0.845098 || 27 | 1.431864 || 47 | 1.672098 || 67 | 1.826075 || 87 | 1.989519 
8 | 0.903090 || 28 | 1.447158 || 48 | 1.681241 || 68 | 1.882509 |; 88 | 1.944483 
9 |, 0.954243 || 29 | 1.462398 || 49 | 1.690196 || 69 | 1.838849 || 89 | 1.949390 
10 | 1.000000 || 30 | 1.477121 || 50 | 1.698970 || 70 | 1.845098 |; 90 | 1.954243 
11 | 1.041893 || 31 | 1.491362 || 51 | 1.707570 || 71 | 1.851258 || 91 | 1.959041 
12 | 1.079181 || 32} 1.505150 || 52 | 1.716003 || 72 | 1.857332 || 92 | 1.963788 
13 1.113943 || 83 | 1.518514 || 53 | 1.724276 || 73 1.863323 || 93 | 1.968483 
14 | 1.146128 || 34 | 1.531479 || 54 | 1.782894 || 74 | 1.869282 || 94 | 1.973128 
15 | 1.176091 || 35 | 1.544068 || 55 | 1.740363 || 75 | 1.875061 || 95 | 1.977724 
16 | 1.204120 || 36 | 1.556303 || 56 | 1.748188 || 76 | 1.880814 || 96 | 1.98227 
17 | . 1.230449 || 37 | 1.568202 || 57 | 1.755875 i 77 | 1.886491 || 97 | 1.986772 
18 1.255272 38 | 1.579784 || 58 | 1.768428 || 78 | 1.892095 || 98 | 1.991226 
19 1.278754 || 39 | 1.591065 || 59 | 1.770852 || 79 | 1.897627 || 99 | 1.995635 
20 | 1.301030 || 40 | 1.602060 || 60 | 1.778151 || 80 | 1.903090 ||100 | 2.000000 
{ ! 
| Sign Si Val Si Val Si Val 
| a7 Sig r Sign | Value} Sign alue| Sign alue 
Jaton. in ist Veins in 2d at in 3d at j|in4th} at 
Lac Fd Quad. |" *| Quad.| 180°. | Quad.! 270° | Quad.} 360°. 
——— | ae | =| | (Ee 
Binv?. 02 Peep pe ee FMM oes ey |* Ne reed) 
Tan Oo oe) _ O a oa) — O 
Sent PSR R co coc to ts RH ae oO Le R 
Versin....| O aE R + 2R {| + R -- O 
COS tS tacts | R O _ R | — O +. R 
Oban Jee ae |} @ -++- O -- oa +f O ~ 00 
Oosec. ..<.. | io) + R -{- a0 _ R _— ora) 
—— — — — — t —_ — ——$_—___—_— = iy 


R signifies equal to rad; © signifies infinite; O signifies evanescent. | 


308 


60 


120 


180 | 
240 | 
300 | 
360 | 


420 
480 


AO | 
600 
660 | 
720 
780 


840 


900 
960 
1020 | 
1080 
| 1140 


200 


1260 | 
1320 | 22 
1380 | 2% 
1440 | 
1500 | 2 


1560 
1620 
1680 
1740 
1800 
1860 
1920 


1980 | 3: 
2040 | < 
2160 | 3: 
2160 | 3 


2226 
2280 
2340 


2400 | 4( 


2460 


2520 


Ow) | 
2580 | 


2640 


2700 | 4 
2760 | 


2820 
2880 
2940 


3000 | 5 


3060 
3120 
i 3180 


| $240 | 
2300 | ! 


3360 
' 8120 


| 8180 Ie 
| 8340 |! 


3600 


bat or) 


~3 


~ 


CO ~3 


(o-2) 


TABLE XXV.—LOGARITHMIC SINES. 


Sine. 


Inf. neg. 
| 6.463726 
|. .764756 


. 940847 
065786 
. 162696 


241877 | 


. 808824 
.3866816 
.417968 
.463726 


542906 


577668 | 
609853 | 2 


639816 


667845 
694173 | 
718997 
742478 | 


164754 
785943 
.806146 
825451 
843934 
.861662 
878695 
895085 
. 910879 
.926119 
. 940842 


. 955082 
. 968870 
982233 
. 995198 
007787 
.020021 
.031919 
043501 
054781 
065776 


J 


| 8.076500 


086965 
097183 
107167 
. 116926 
126471 
. 135810 
. 144953 
. 153907 
- 162681 
.171280 
179713 
187985 
. 196102 
204070 


.211895 | 556 


219581 
227134 
234557 


| 8.241855 


| 
~N 


Tang. 


1 Q 
2) 


Or 


— O> 


=? 


Ovorororororor 
PO Is oe 
Ovoror or or 
IS 


ry Or Or Or Or Or Or 


Ia 3 I I I 


It 


He ee OT OT OT 


orer Oror1roror 
{I QR 


ha 


505118 | 


tJ 
tS 
~ 


-z 


Cosine. 


f. neg. 
463726 
164756 
. 940847 
.065786 
. 162696 
241878 
808825 
.866817 
.417970 
.463727 
7.505120 
.542909 
577672 
.609857 
639820 
667849 
694179 
719008 
. 742484 
(64761 
7.785951 
.806155 
825460 
843944 
.861674 
878708 
895099 
910894 
926134 
. 940858 


. 955100 
. 968889 
982253 
.995219 
007809 
-020044 
.031945 
048527 
.054809 
.065806 


95 8.076531 

3} 086997 
097217 
107208 
116963 
126510 
135851 
144996 
153952 
162727 


171328 
.179763 
. 188036 
. 196156 
204126 
-211953 
219641 
220195 
.234621 
8.241921 


Cotang. 


13.536274 
235244 
13.059153 
12.934214 
837304 
758122 
691175 
633183 
582030 
53627 
12.494880 
457091 
422398 
390143 
. 360180 
332151 
305821 
280997 
257516 
235239 
12.214049 
193845 
174540 
“156056 
138326 
121292 
104901 
089106 
073866 
059142 
12.044900 
031111 
017747 


12.004781 


. 979956 
. 968055 
.956473 
.945191 
. 984194 
. 923469 
. 913003 
902783 
892797 


883037 


1 


are 


.864149 
. 855604 
. 846048 
837273 
.828672 
~ 820237 
.811964 
. 803844 
795874 
. 788047 
780359 
.77 2805 
. 765379 


11.758079 


1 


are 


Tang. 


Cotang. 


Inf. pos. | 


We) 


=) 


ive} 


| 11.992191 


ie) 


.873490 


Sine: 


ID 1" Cosine, 


ten 
ten 
ten 
ten 
ten 
ten 


. 999999 | f 
. 999999 
. 999999 
. 999999 
. 999998 


999998 
999997 
-999997 | 
999996 

999996 
999995 
999995. | 
“999994 
“999993 
999993 


9.999992 | 
-999991 |" 
. 999990 
. 999989 
. 999989 
. 999988 
.999987 
. 999986 
. 999985 
. 999983 


.999982 
. 999981 
.999980 
.999979 
989977 
.999976 
.999975 
. 999973 
. 999972 
.999971 


. 999969 
. 999968 
. 999966 
999964. 
. 999963 
. 999961 
999959 
. 999958 
999956 | 
. 999954 
| 
999952 
. 999950 
999948 
. 999946 
. 899944 
. 999942 
. 999940 
. 999938 


999936 


-03 | 9’ 999934 


3960 
4020 
4080 
4140 
4200 


4260 
4320 
4380 
4440 
4500 
4560 
4620 
4680 
4740 
4800 


4860 
4920 
4980 
5040 
5100 
5160 
5220 
5280 
5340 
5400 
5460 
5520 
5580 
5640 
5700 
5760 
5820 
5880 
5940 
6000 


6060 
6120 
6180 
6240 
6300 
6360 
6420 
6480 
6540 
6600 


| 
6660 


6720 
6780 
6840 
6900 
6960 
7020 
7080 
7140 
7200 


4 Sine. q—t 
| 4.685 
0 | 8.241855 | 553 | 619 
1) .249033 | 552 | 620 | 
2) 256094 | 551 | 622 
3 | 263042 | 551 | 623 | 
4 | .269881 | 550 ,| 625 | 
5 | .276614 | 549 | 627 
6 | 288243 | 548 | 628 
7 | ».289778 | 547 || 630| 
8 | .296207 | 546 | 632 
9 | .302546 | 546 | 633 | 
10 | .308794 | 545 :| 635 | 
11 | 8.314954 | 544 | 637 
12.| 321027 | 543 | 638 
13 | .827016 | 542 | 640 
14| .332924 | 541 | 642 
15| .338753 | 540 | 644 
16| .844504 | 5389 | 646 
* | 850181 | 589. | 648 
18 | .355783 | 588 || 649 
19 | .3613815 | 5387.,| 651 
20 | .366777 | 536 || 653 
21 | 8.372171 | 585 | 655 
22 | .377499 | 534 | 657 
23 | .382762 | 533 || 659 
24 | .387962 | 532 | 661 
25 | .893101 | 531 | 663 
26 | .398179 | 530 | 666 
27 | .403199 | 529 | 668 
28 | .408161 | 527 | 670 
29 | .413068 | 526 | 672 
30) 417919 | 525, 67 
31 | 8.422717 | 524 || 67 
32 | .427462 | 528 || 67 
33 | .482156 | 522 || 681 
34.| 436800 | 521 || 683 
35 | 441894 | 520 || 685 
36 | .445941 | 518 || 688 
87 | .450440 | 517 || 690 | 
38. | .454898 | 516 | 693 
39 | .459301 | 515 || 695 
40 | .463665 | 514 | 697 
41 | 8.467985 | 512 || 700 
42 | .472263 | 511 || 702 
43 |  .476498 | 510 || 705 
44 | .480693 | 509 || 707 
45 | .484848 | 507 || 710 
46 |. .488963 | 506 || 713 
47 | .493040 | 505 || 715 
48 | .497078 | 503 || 718 
49 | 501080 | 502 | 720 
50 | .505045 | 501 || 723 
51 | 8.508974 | 499 | 726 
52 | 512867 | 498 | 729 
58 | .516726 | 497 || 731 
54. | .520551 | 495 || 734 
55 | | 524343 | 494 || 737 
56 | .528102 | 492 740 
57 | .531828 | 491 || 748 
58 | .5355283 | 490 | 745 
59 | .539186 | 488 | 748. 
60 | 8.542819 | 487) | 751 
4.685 
’ | Cosine. q—l 


TABLE XXV.—LOGARITHMIC SINES, 


Tang. 


8.241921 
249102 
.256165 
.263115 
. 269956 
-276691 
283323 
289856 
296292 
802634 
. 808884 


| 8.315046 


-321122 
.827114 
833025 
. 388856 
.844610 
850289 
800895 
.861430 
. 866895 


8.372292 
3877622 
882889 
. 888092 
38932 
3898315 
403838 
-408304 
413218 
.418068 


8.422869 
427618 
432315 
-486962 
-441560 
.446110 
-450618 
455070 
-459481 
463849 


8.468172 
472454 
476693 
480892 
485050 
489170 
493250 
497293 
501298 
505267 

8.509200 
513098 
516961 
520790 

524586 
528349 
532080 
535779 
539447 

8.543084 


11. 758079 
750898 
748885 
. 736885 
730044 
.(23309 
16677 
710144 
103708 
. 6973866 
.691116 


684954 
678878 
. 672886 
666975 
.661144 
.655390 
649711 
.644105 
688570 
633105 
. 627708 
622378 
617111 
.611908 
. 606766 
601685 
.596662 
.591696 
586787 
581982 


577181 
572882 
. 567685 
.568038 
558440 
.558590 
.549887 


.544980 


11 


11 


11 


.586151 
11 
027546 
.528807 
.519108 
.514950 
.510880 


498702 


1 


bak 


.486902 
.488039 
.479210 
475414 
.471651 
.467920 
.464221 
.460553 
11.456916 


Tang. 


Cotang. 


.540519 | 


581828 


.£06750 


502707 | 


494733 
.490800 | 


| 


gti ie at | 
15,314'| | [ere 
381 |! gg | 9.999934 | 60 
380 | ‘gs 999982 | 59 
378 | "pg  -999929 | 5 
ce | "03 .999927 | 57 
75 |) "on .999925 | 56 | 
73 || "ga | .999922 | 55 
72 || "FS | 999920 | 54 
70 | “ox | .999918 | 53 
368 “Og 999915 | 52 
867 | ‘os | -999913 | 51 
365 || “~? | ..999910 | 50 
363 | “03 9.999907 | 49 
362 | "os | - -999905 | 48 
860 | ‘op . -999902 | 47 | 
358 “og — .999899 | 46 
856 ‘ps, -999897 | 45 
854 "on | .999804 | 44 
852 ‘pp | .999801 | 43 
851 | “pe |. 999888 | 42 
3849 | ‘on | .999885 | 41 
847 |" 9.999882 | 40 
345 | “0? 9.999879 | 39 
343 “gs | .999876 | 38 
841 || ‘92 | .999873 | 37 
839 | “op |. .999870 | 36 
B37 | "op | .999867 | 85 
834 | ‘or |  .999864 | 34 
832 | “oe | .999861 | 33 
830 |/.‘97 | .999858 | 82 
328 || “os 999854 | 31 
326 || ° .999851 | 30 
aed | +03] 9.999848 | 29 
321 “05 999844 | 28 
819 || "op | .999841 | 27 
817 || “p» | -999888 | 26 
| 815 |] ‘os | -999884 | 25 
312 || 72 | .999831 | 24 
810 | 05 999827 | 23 
B07 |) “ow | . 999824 | 22 
| 805 || ‘o | . 999820 | 21 
303 |} °** | . .999816 | 20 
| 300 || 03 | 9.990813 | 19 
| 298 || a. | ..999809 | 18 
| 295 |) "oy |. 999805 | 17 
| 293 || ‘oy | . 999801 | 16 
290 || or | .. 969797 | 15 
287 Il ae]. .999794 | 14 
285 || “g» | .999790 | 18 
282 || “G, | . 999786 | 12 
280 | -o | ..999782 | 11 
| 277 |)" | 999778 | 10 
ar Eis 9.999774 | 9 
271 | ‘on | 999769 | 8 
269 || “oy | ..999765 | 7 
266 || ‘oy | ..999761 | 6 
263 || a4) ..999757 | 5 
266 “OR | ..9997538 | 4 
257 || “ay | ..999748 | 38 
255 || 741 ..999744 | -2 
252 || ‘og | ..999740 | 1 
249 y |) 9.999735 | 0 
15.314| 
| 
wey Peay PERMA Levee 
q+1||D1*| Sine. [aes 


Dale. 


! 
4 Sine. Dek. Cosine. | D. 1’. Tang. 
0 | 8 542819 9.999735 | 8.543094 
1 | .546422 ote | ‘999731 | “Oy || .546691 
2 .549995 59. 07 . 999726 OF. Web 550268 
3 | .553539 | 59" 5g || -999722 08 553817 
4| .557054 | se'y9 || -999717 07 557336 
5 | 560540 | 2 ae || .999718 “08 560828 
6 | .563999 | 24-09 || 999708 | ‘oe 564291 
7 | ‘pezasr | 26-22 |) 999704 | “og 567727 
8 | (570886 | 26-49 || 999699 | og || -571187 
| ipra2i4 | 56:39 || “999604 | “og || 574520 
10 577566 | 58:82 || (999680 | gy || -577877 
11 | 8.580892 | ~» no || 9.999685 og || 8.581208 
12| .584193 | 24'¢9 || -999680 08 584514 
13 | .587469 | Pa'o9 || -9996%5 “08 587795 
14| .59072t | Za'ng || -999670 “Og 591051 
15 | .593948 | P3749 || -999665 | “og 594283 
16| .597152 | > “00 . 999660 “08 597492 
17 | 7600332 | 23-09 || ‘999655 | “og || .600677 
18 | 603489 | 52-08 || 999650 | ‘og ||. 603889 
19 | .60662% =. || 999645 “68 606978 
20 | ‘609734 | 51-85 |) “900640 | gg |] | -610004 
21 | 8.612823 | ~ | 9.999635 8.613189 
S| 615801 | 51-8 || ~ ‘990629 40 || “e1626e 
23 | .618937 ce “ho || 999624 “08 619313 
24 | .621962 | sng || -999619 “08 622343 
25 | .624965 | Gong || -999614 | “49 . 625352 
26 | .627948 | 4o'ag || -999608 | ‘og 628340 
27 | .630911 | 4o‘g5 || 299603 <6 631308 
28 | .633854 48.70 999597 08 634256 
29 | 636776 “ty || -999592 | “40 637184 
30 | 639680 | 48-40 || ‘909586 | “og. || - 640098 
31 | 8.642563 | ye we || 9.999581 | 8.642982 
a2} 645428 | 40-12 || .9995%5 “40. || 645858 
83} .648274 | “4eyg || -99957 "46 648704 
84| 651102 | 4e'gs || -999564 40 651537 
35 | .653911 | Je '55 || 999558 6 654352 
36 | .656702 | Je'o9 || -999553 46 657149 
387 | .659475 | Je‘go || -999547 10 659928 
38-| .662230 | 4p'gg || 999541 “40 .662689 
39 | .664968 0-09 |) 999535 30 665433 
40 | ‘667689 | 45-25 || 999529 | og || -668160 
41 | 8.670893 | 4, -a || 9.999524 8.670870 
42 | .673080 | at || 999518 “10° || 2673568 
43 | 675751 | 4493 || -999512 “10 676239 
44| .678405 | 43’g7 ||  -999506 16 678900 
45 | .681043 | 43mg || -999500 es 681544 
48 | 683665 | 43°45 || 999493 a3 684172 
7 | .686272 | 13.18 | 999487 10 686784 
48 | 688863 | 45%o9 || -999481 40 689381 
49 | 691438 | 45"pe || -9994% “10 691963 
50 | .693998 | 45" 4o || -999469 i0 694529 | 
B1 |. 8.696543 |< 9.999468 8.697081 
52 | (699073 | 42-44 || 990456 22 || 1699617 
53 | 701589 | 41 7¢g || 999450 12 702139 
54] 704090 | 45 74x || -999448 10 7104646 
Bs | .706577 | 4y‘o9 || -999487 | “10 107140 
56 | .709049 | dog || -999431 49 . 709618 
7 | .711507 | go's || -999424 45 712088 
58 | .713952 | 40 53 .999418 "yo. (|| 14584 
59 | .716383 | 4n'9g || - 999411 19 || ,- 716072 
60 | 8.718800 | 42-8 || 9.999404 2 || 8.719396 
fi Cos*~e. | D1”. |i. Sine. Dat Cotang. 


ee 


Cotang. | f 
| 


11.456916 | 


453309 | 59 
.449732 | 58 
-446183 | 57 
.442664 | 56 
.489172 | 55 
.485709 | 54 
.482278 | 53 
428863 | 52 
.425480 | 51 
.422123 | 50 
11.418792 | 49 
.415486 | 48 
.412205 | 47 
.408949 | 46 
405717 | 45 
"402508 | 44 
.899823 | 438 
.896161 | 42 
.398022 | 41 
889906 | 40 
11.386811 | 39 
.883738 | 38 
.880687 | 37 
.877657 | 36 
.874648 | 35 
.3871660 | 34 
.868692 | 33 
.3865744 | 382 
.862816 | 31 
.3809907 | 30 
11.357018 | 29 
.854147 | 28 
.3891296 | 27 
348463 | 26 
.345648 | 25 
. 842851 | 24 
.840072 | 23 
.8dlell | 22 
.034567 | 21 
331840 | 20 
11.329180 | 19 
.826487 | 18 
823761 7 
.821100 | 16 
"318456 | 15 
.815828 | 14 
.313216 | 18 
.3810619 z 
.8080387 | 11 
.205471 | 10 
11.302919 9 
. 800383 8 
297861 cs 
. 295354 6 
. 292860 5 
.290882 | 4 
287917 | 3 | 
(285466 | 2 | 
"283028 | 1 | 
11.280604 | 0 
Tange 3 


87° 


| 


| 


Cosine. | D. 1’. 


PP ELAS « 


| 
g fotanzs | 
Tang. Didi © g | 
Sine. Ds We: ft 
| per of ieee 11.280604 | 69 
| ges VW |" “O7gi94 | 5 
; 7 || 9-999404 | 49 || 8 721806 eS oy | <2iS1ML 09 
0 | 8.718800 | 40.0% .999398 "19 | oe 29.9% i 3 
1 | .721204 | 90° e6 999391 13: || | ete 20.73 Bate | 
2 | .423595 | 39 69 “999384 “35 728088 20.52 | Panes 
Pe load ee Tre -999878 | “15 728059 39.30 le: 
4} .728337 39.18 || “go93°4 12 7aa663 | 39-10| 08s 
‘ 2878 BO8887 12 735996 38.68 | "961683 | 52 
ic) visas | ‘Boon | ie BBY | Sotelo 259374 | Bl 
| Bat) Ba ‘gua | 2 740626 | 38°57 "957078 | 50 
gal Dipsgoen | Bae 999348 eS | ae 
10'|  fraeesa | Son! nee e Sasa ae eae 
"999; 12 | ravaro | 3087 Be 48 
11 | 8.744536 | ge gggaee | +12 ep | Bi . 
2 - (46802 37.55 999315 aR fare 7.48 | peu ‘a 
13 | .749055 | 3°3> . 999308 39 edie 3730 28 | 
14] .151297 | eg 999301 | “12 Tee | 87.10 pag 
dott Udamcone 1320308 Sees hasta ||| + eae so) cause | 
16 | 55747 86.80 ||  “ gq999% by a | 8.7 Bis i 
17 U5T955 36.60 999279 “12 fosois | 28-25 Ss | 4 
18 | 760151 36 43 fond ee ‘ants a8 | | 
202) | WEASEL | Bee dl | - ; : cma | 2 ss | 
a od=4 9.999257 12 e058 a ane : : 
21 | 8.706875 | 55 9g 999250 | 22 fu | | Bat ei 
BE) OBO | BR ‘goa | 18 Tt | 208 
a imo ann || See ees -€05995 | 935" 39 "991886 | 34 
baal wecags (> BUIBRS | eee iB | Fae Boe: 
25) .775223 | ae "ye 999220 || es | 35.38 ie: 
26 | .777333 | 9x" 02 999212 "12 eee 3497 21th : 
ign, if Nate os : = j Savane “4a ‘feuos | 2480 216 3 
mR 36C : .999197 ‘ “ 3 34.63 > 
: weber, | 24:50 999189 — 786490 tu eee 
30 | {785675 coment it com) he aap 
‘30e Pooorra | 22. |] PFapets 3415 | 200887 | 28 
mie | 38 ‘soos | 28 || “toeg62 33.98 | “Sos999 | 96 
Ss | tomes | 0 |) poo ee || | zed | apie 203269 | 25 
x rss 33.70 te 13 ; (96031 33.68 "001248 24 
sr | Bag | gun AD || | goeene | Bee 199237 | 28 
6 : 804 33.38 oa 13 || 800763 33.37 "197235 | 22 
aq) aie | a |) | Boeees | ears 195242 | 21 
88 | .801892 | 33 o7 rch io 13 wie | Big) oi 
39 803876 | 82.93 “999110 a ‘$2 | ee : 
ass | 3 | Sa | spay 89317 | 18 
40] | = : : ; : 
ie vant | | ‘Bipoa | 32-68 187359 | 17 
41 | 8.807819 | 99 ¢3 999004 | “43 1008 | 22.08 a | 
42 809777 32.48 “999086 i. eel | eat i | 
43 811726 32.35 .999077 13 ‘lame | 233 tera | 15 
44} 813667 32.20 999069 13 sisioi | 32.20 1815 | 1 
45 ‘ 815599 82.05 999061 “43 ; yee s 05 ra c 
46 | .817522 | 31°99 ee 13 0 | 58 ms 
47 | 819436 | 3) "ng 999044 43 sims | L78 135 1 
49 | ‘ss3040 | 31-62 || “po9034 a5 |) Se | ares) 
49 | 828240 | 355 999086 i * a aa : 
50] j625i80 | Sten || |: . comm | os ras 
ai ri 9.999019 | 4. Oey atan | fam | 3 
RE OE en eget tts 8 |) Seg] aa] i 
oe) eet | arenas, eee 15 | “33613 | 31:08 160387 | 4 
pt | ‘sscor | 32-82) “ooeags 1h. |) garam | Bhar “360670 | 4 
BS | isouise | 30-82 || ones ieee Sa 160837 | 3 
BS) | 3s “goss a 839163 30.58 "159002 | 2 
pr | isgsizo | 30-55 . 998967 15 || “g4g98 30.45 "157175 | 1 
me ese | 3) || ankton |. Sapam. |S oete 1:155356 | 0 
Bea eevee |" Bree 998950 | Te Wri Bay 1 
col Ripiages 4¢: B18 9.998941 | 8.844644 by vo 3 
7 | Sine. D. 1”. || Cotang. | D. 1 é 
i | D 1" Sine. 5 ae. 
‘~} Cosine. 


COSINES, TANGENTS, AND COTANGENTS. 


iar 
’ | Sine. /D, ae Cosine. | D. 1". Tang. D. 1". | Cotang. Oy 


| 


| Se a 
| 11.155356 | 60 


| 8.843585 | , || 9.998941 8.844644 


3751 ©, 905736 | 52: .092853 | 23 
38 | .907297 | *&?- 
39 | .908853 | 2: 
| 910404 


41 | 8.911949 
2 . 913488 
43 . 915022 
44 | .916550 
45 . 918073 


998589 
998578 
“998568 
-©2 |) 998558 


907147 | $2: 
"908719 i 091281 | 22 
910285 | Se'go | . 080715 | 21 
-911846 | S5"99 | 088154 | 20 


© 
<= 
Di 


Oro 
(9 2) 
or 


~~ 
| 0 | 8.848585 | 35 og tes 15 || S-BasOe* | e998 
1 api | 29.93 || -998982 "15 || -846455 | g'9g | — .158545 | 59 
Btemerano |OROS00 Ihccapsee, | OLAS It) eee Minton | epdet ad 158 
3 | .848971 | SoG || .998014 15 |; -850057 | So'go | -149948 | 5% 
4) .850751 | 99'5> || -998905 | “48 851846 | Sony | -148154 | 56 
5 | -R5w25 | a9 43 |) 908606 | 75 | -soBoNs | DOS | 14632 | 5b 
| &) S549) Soca5 || -2abesz | 18 || -soed03 | Soh | 14s5a7 | 54 
g} -c20he) | 99.20 || 22888 | a5 || 857171 | -Oo°g5 | — .142829 | 53 
j 8 | .857801°'| 5998 .998869 45 858932 | $9’ o9 141068 | 52 
‘a sarees | 28 95 eae "45 860686 | 9942 |  -189814 | 51 
| O85 | tae || O98R5 a eke ENS a6 wd.in | errer 
| a 28-85 || spe | Mea 862433 99/09 |  - 180564 | 50 
11 | 8.863014 | 96 7g || 9.998841 15 || 8.864173 | og ge | 11.185827 | 49 
a | Leny Hos | 98°69 || fod ae 865906 poor .184094 | 48 
t ¢ Oe Ty || . B2¢ Pe 567635 | ao 3236 | 47 Wi 
14| ‘g68ie5 | 28-50 || ‘ooggi3 | -17 "860851 eres Lae 46 
ae Eececonés | 228928 \rseacaod | oI. |bateermes |obeeb5 | norapeees p86 
Die cmtan | 282R8 jlosiegaron | 15. Par gigoee | Mg as |) .xfeeese fae 
16 | 871565 | 98°5~ 998795 0 82770 | Saag | 127280 | 44 
: 17 | 878255 | So'oe ||. 998785 er 874469 | So°35 | .125531 | 43 
18 | 874938 28.05 | C0876 Sait 4 Ly 28.22 | 99e< | 
| mv | eects 2795 || easoag 17 eee 98 19 . 123838 42 
: 9 | 2O10615 | nee ee | .IIO(6 cans R7784! Prien OTT | 
20 | ie7eess | 27-83 || ‘gggtsz | 15 |] “erasag | 28:00) -Fourt | Go 
i oscs, DON ace Sas | Rea aaa se ek 
21 | 8.879949 | oe po || 9.998747 ~ || 8.881202 | og» we | 11.118798 | 39 
: 22 | 881607 | Spo || 908738 | 13 || “gence | 20-78 | "117131 | 38 | 
: 23 | 888258 | 92745 || 998728 , 884530 | Soke .115470 | 37 i 
24 | 884903 | So°gq || .998718 | 4 886185 | Ger |  .118815 | 36 
Bees near | REPO loi eeeee | A |i, BREE Abe as | TIMtOT 85 t 
| 26 | 8881 (4) oe Fo . 998699 er 289476 oy on |  .110524 | 34 i 
BE Pee oy) SPO liigieeegey | O17 llaxmeanien Coed? | tutes dees 
28 | .891421 | S445 |] .99867% a 892742) | Sy 7258 | 3: 
99 | 893035 | 26.90 || ae i pithas 27.0% 107258 32 i 
Gr tawinsers | 2000 ockenecc:| Celt |feadteence | 26-07 | er ckeeOes fel i 
| 80 | 894643 | 36 79 || 998655 ty -895984 | Se g~ |  .104016 | 30 
B1 | 8'896246 | 96 gg || 9.998649 | 17 || 8.897596 | 96 ng | 1.102404 | 29 | 
82 | 897842 | Seiry ||  -998639 1” -899203 | $6'gy |  -100797 | 28 
| ss 890432 | 26.42 || -998629 | “yy .900803 | Seng |  -099197 | 27 
$ ( t | aU.26 |! R6 . QN9202 wd. led Vays a 
Be ratiercs 020-82 les taeceie 1 Sci? |lesneesees | oe'as | on Uez60e y26 
De Mahecrics | P2022 Ih wlgueeon) | Tale: llestareven | 2008S | ryepnaeee ieee 
36 904169 | 56°45 . 998595 47 . 905570 9698 .094430 | 24 
18 
17 
17 
1? 
rf 


. 918401 


= 
=) 


65 || 9-998548 OF 83 | 
ar || 998537 914951 | Orme | 085049 | 18 
“47 | 998527 |} -916495 | 95° no |  .088505 | 17 
‘oo ||. 998516 BS “018034 | aoa 081966 | 16 
; 998506 wos 919568 | S2°°S | 080432 | 15 


gg | 11.086599 


ID 
co 


TOL OU OL OTOL 
iY) 
CO 


OW WWW PW IW WW 


_ 
° 


46 | 919591 | $59 || -998495 | “yn -921096 | Ge°ag |  -078904 | 14 

47 | .921103 | G2 "45 || 998485 | “42 -922619'| Sr "So 077881 | 13° | | 

48 | .922610 | orgs || -998474 | 4 924136 | Se°So | 075864 | 12 | | 
| 49 | .924112 | Oy'gn ||  .998464 "ig || -925649 | eo. | eed! ay aes 


998453 QRTIEB.| “SecRs .072844 | 10 
9.998442 1g || 8-926658 | x | 11.071342 
. 52 928587 | 37: SOBABT | ates tS */O80155 1) Agata bi SY OG9845 

53 | .930068 | Sigg || 998421 | Hl 981647 | os ates 068353 
54 | 931544 | O)'r5 || .998410 a8 933134 066866 
a | » 1 


5U .925609 | 9S," 


51 | 8.927100} 


ra) 
x 
SO 
Qo 


55 | .933015 | S¢-28 998399 | ; 984616 | S7rnc 065884 

=o be ps 24.43 boa | tS Anas 4 eae ohne 

56 | .934481 | Siia~ || .998888 | : . 986093 | iS 063907 

By | 935942 | 24.35 | 998307 18 02" C5 | 24. 162435 

of JOIAIAK 2 27 j JI | 18 ~JO(O0D | 94 45 002435 
| | 


58 | .987398 || .998866 . 9850382. | 


O49 060968 


| 94,909 | | |), Aes 4 
59 | 938850 | 31:9? || ‘998355 | 48 || ‘940494 Sr ay |... .059506 
60 | 8.940206 | **:10 || 9 oogaa4 | - 8.941952 | “4-9 | 14 "o58048 
| 


‘ | Cosine. | D. 1’. 


Sine. | D.1’. || Cotang.| D.1". | Tang. 


co ~| COR wwhonwMHO.e 
qn | 
° 


94° 


TABLE XXV.—LOGARITHMIC. SINES, 


3 Sine. D. 1", || Cosine, | D. 1’. 
ene: ARON See Desay 
| pares | 2403 | ogesss | 18 
2| 943174 | 23°25 || .gogsee | -48 
€ A ane | we. | O82 oA 
4} “p1o3t | 23-80 |) “goss00 | 18 
| MARC wO.dU | QgR9R 16 
6 | coseera | 23-63 || ‘gepaey | 20 
7 | 1950287 eae || -pgeece | trie 
FRED ae teed) | Giant | fe Bp 
9 | 1953100 | 33-29 || ‘oge24: By 
10 | 954499 | 53-82 || ‘oogzae | -28 
11 | 8.955804 | 45 "4p 9.998220 ie 
‘ 9575 3. QgR9(x : 
13 | lasso70 | 23-10) “ggsior | 20 
14 | 960052 | $3°p5 || -298185 | “2p 
Mae 2.87 cs lle .18 
17 | -oeivo | 22-82 || copsist | 2° 
18 | 965534 | 99°65 || 998189] “4g 
19 | 966803 | 22-65 || ‘oogi28.|  -38 
20 | .968249 | $3-89 || ‘oosiie | 30 
21 | 8.969600 | 5 yx || 9.998104 m5 
i- rngan | reed 992099 . 
53 | ‘o7e089 | 22-37 || “booq | -20 
24 | 973628 | 22-82 |] “gogogg | -20 
25 | 974962 | 22-23 |) ‘oggos6 | -20 
Be eens | Hoed0 [I beoaananll| Re 20 
35 | coreoat | 22-03 || “gogpnn | -20 
20 | 980259 | 21-97 || ‘oogoo8 | -22 
30 | .981573 | 31:39 || .997006 | «30 
98288; rw || 9.99798 
32 | cossisg | 24-27 || “gorore | -20 
Qe FAQ @lwla oar BC oe 
at | ‘gsorsa | 21-63 |] “gyrauy | 20 
35 | 988083 | 9150 |) -907085 | op 
37 | 990660 | 21-43 || “ggroio | -20 
38 | 1991943 | 21-88 || ‘ogegor | -22 
39 | 993222 13a. || eacoenesb || ate 
40 | .994497 a ie || 907B72,| S25, 
41 | 8.995768 | 5, 4» || 9.997860 ae 
¢ 907028 ra e 9978 ~ «ee 
43 | vgog209 | 21-05 |] “Boreas | 20 
44 | 8.999560 | 21-02 || “gg7gog | +22 
45 | 9.000816 | 20:93 || ‘gore | -22 
lee epee (ereo288 ll aaenaecoy | mee 
ae eaanara (re20-82 ete 18422 
48 | 1004563 | 20-75 || “ggvmay |  .22 
49 | 005805 ores | lgorr5a |  -22 
50 | .007044 | 55° 57 | 997745 39 
51 | 9.008278 | | 9.997732 
2 20.53 cit 22 
52 1.009510 | 99" 45 oe | aaeee 
BP Ve tetany [ede (a eeaens | base 
bs | loisiea | 20-83 || “orega | 22 
56 | .014400 | 20-80 || “oovee7 | -22 
»& =< € ae | ner 7 
pe oMaiercese | 2018 || Seed pee 
BoP hiere, (20d? aeeeent | Maps 
60 | 9.019235 | 9.997614 
4 Cosine. | D. 1" Dsl" 


Sine. 


174° 


Tang. D. 1". | Cotang. | ’ | 
| 8.941952 | 94 99 | 11.058048 | 60 
943404 | Sits .056596 | 59 
|| -944852 | Sips | -055148.| 58 
|| -946295 | 93°98 053705 | 57 
leew 947734 oS “90 052266 | 56 
| 949168 | 55° o5 "050832 | 55 
950597 | G3" mn .049403 | 54 
952021 | 95" 67 047979 | 53 
953441 | 53"59 |  -046559 | 52 
954856 | 99'55-|  -045144 | 51 
956267 | 93" 4r 043733 | 50 
| 8.957674 | oe a | 11.042326 | 49 
| .959075 aah "040925 | 48 
|| -960473 | $355 039527 | 47 
-961866 | $9°T5 .038134 | 46 
. 963255 93 07 086745 | 45 
964639 93 00 .035361 | 44 
.966019 99 “99 .033981 | 43 
967394 | 55" ge 032606 | 42 
968766 ‘ay 2 031234 | 44 
970133 99.79 .029867 | 40 
8.971496 | 55 gx | 11.028504 | 39 
1972855 | es 0 027145 | 38 
-974209 | 55°29 .025791 | 37 
975560 | 55°43 .024440 | 36 
-976906 | 99’ 30 . 023094 | 35 
978248 | 55°39 021752 | 3 
979586. | SS-Se 020414 | 33 
-986921 | 55°4r .019079 | 32 
982251 | 95° 0 017749 | 31 
983577 203 .016423 | 30 
8.984899 | 9, gy | 11.015101 | 29 
.986217 | 51 "99 | 018783 | 28 
987532 | 51 gs 012468 | 97 
. 988842 | 5; ne .011158 | 26 
-990149 | 54", 009851 | 25 
991451 | 9) 65 008549 } 24 
992750 | 61 "58 007250 | 23 
994045 | 94 "R3 005935 | 22 
995337 | 94°45 .004663 | 2 
996624 | 54°45 003376 | 20 
8.997908 | 94 99 | 11.002092 | 19 
8.999188 | 9,58 | 11.000812 | 18 
| 9.000465 | 94°59 | 10.999535 | 17 
001788 | 94°45 .998262 | 16 
-003007 | 519g 996993 | 15 
004272 | 51°93 995728 | 14 
005584 | “Sa°e8 .994466 | 13 
-006792 | 30"¢ Mi .993208 | 12 
-008047 | 55 gx .991953 | 114 
-009298 | 59°gq .990702 | 10 
9.010546 | 99 we | 10.989454 | 9 
-011790 | 50‘ 68 . 988210 | 8 
-013031 | 59° go 986969 | 7 
-014268 | 5p ‘ae 985732 | 6 
015502 30) 50 .984498 | 5 
016732 | 9945 983268 | 4 
-017959 | 59°49 .982041 | 3 
-019183 | $9‘33 .980817 | 2 
-020403 | 59 ‘g 979597 | 1 
9.021620 *“° | 10.978380 | 0 
Cotang. | D. 1’ Tang. d 


\ 


oo 
a 


COSINES, TANGENTS, AND COTANGENTS. 


Cotang. 


911902 


| 10.910856 


Sine. | D.1". || Cosine. | Tang. 
| 
#019285 | 99.00 || 9-997614 | 9.021620 | 
020435 | 39 ‘95 . 997601 022834 
-021632 | 49" g9 997588 024044 
-022825 | 49 gs 990574 .025251 
-024016 | 397m 997561 026455 
-025203 | 39/0 997547 027655 
-026386 | 39°49 || . 997534 028852 
027567 | 19.62 997520 | .030046 
O28744-| 39" 5> 997507 | .031237 | 
.029918 19 52 . 997493 | 032425 
-031089 | 49 47 997480 | .0383609 
QOOrM | ree OUP 
rary panto. | 097 06 Neekin 
- me ‘Ad | 19.35 ee th ow 085969 | 
| -084582 | 39°90 .997439 .037144 | 
| -085741 | 49'ox || 997425 .038316 | 
| .036896 | 59 ‘5p 997411 .089485 
| .038048 | 39°45 .997897 .040651 
| 039197 | 49'o8 997383 || .041813 
| .040342) 39°p5 || .997369 |} .042973 | 
-041485 ] 49°99 || -997355 || .044130 | 
-042625 | 49’ 95 || .997341 | 045284. | 
9.043762 | 4 g¢ || 9.997327 9.046434 | 
044895 | 38° gx .997313 | .047582 
.046026 18.80 997299 048727 | 
| 047154 | 53h. 997285 || 049869 | 
| -048279 | jo' 63 || .997271 | .051008 | 
-049400 | 39'e5 || .997257 |} .052144 
-050519 | 38° @q || .997242 [| 053277 
051635 | je’ 5m |) .997228 || 054407 | 
052749 | 3Q°59 || 997214 || .055535 
-058859 | 39°45 || .997199 | .056659 
91054966 | 49 49 || 9.997185 || 9.057781 | 
056071 | 18.35 .997170 “69 ||. - 058900 
057172} 38°35 997156 “ox || 060016 | 
-O58271 | 42°57 997141 con {t= eobd Le 
| .059367 | je°So .997127 Sor? || 2062240 
| 060460 | 48°58 997112 "93. «|| ~~ - 068348 
| .061551 | 36°43 .997098 ee .064453 
| -062639 | j6'og || -997083 Seb? .065556 
063724 | 49°93 || .997068 | “5? 066655 
| 064806 | jr'99 || .997053 93 067752 
9.065885 | 17.95 || 9.997039 | 95, || 9.068846 
| 066962 wn 90 || . 997024 On .069938 
| 068056 | je"p5 || .997009 ase 071027 
| 069107 | jn" 25 . 996994 : 072113 
OVO176 | sme || 996979 | 073197 
OT12A2 | sn 'ns . 996964 | .074278 
072306 | 44" 62 . 996949 | .075856 
.073366 63 || 996934 | .076432 
cen 7 60 || -996919 | 077505 
07548 ee :996904 078576 
(.00 
9.076533 750 || 9.996889 9.079644 
077583 | je ge |) .996874 | 080710 | 
.078631 LE pil .996858 08177 
079676 1738 || .996843 | 082833 
| -080719 | 57°39 || .996828 | .083891 
| 081759 | 1730 996812 | .084947 
| 082797 | anos || .996797 |; .086000 
083832 | 44°95 || .996782 |} 087050 
| 084864 | ye'Fy || .996766 .088098 
| 9.085894 | -~‘:** || 9.996751 | 9.089144 
4 | 
Gositre. | Dr1% Sine. || Cotang. | 


Tang. 


| 10.978380 

.977166 

975956 
974749 
973545 
972345 
.971148 
969954 
968763 
967575 |! 
966391 
965209 
.964031 
. 962856 
. 961684 
.960515 
959349 
.958187 
957027 
955870 
954716 


10.953566 
.952418 
. 951273 
. 9501381 
. 948992 
. 947556. 
. 946723 
. 945593 
. 944465 
. 943341 


942219 
. 941100 
. 939984 
. 938870 
. 937760 
. 936652 
985547 
- 934444 
. 938345 
- 932248 


.931154 
. 980062 
928978 
927887 
. 926803 
925722 
924644 
923568 
922495 
921424 


. 920356 
. 919290 
918227 
. 917167 
.916109 | 
. 915058 
. 914000 

912950 


CHUWWRORMUIDS 


~ 


TABLE XXV.—LOGARITHMIC SINES, 


Ro 


co 


| 9.995753 


4 Sine. 
0 | 9.085894 ” 13 
1 | .086922 a 3 
21 087947 tee 
3 | .088970 be 0 
4 | 039990 | Te-p) 
5 | (091008 16 93 
6 | .092024 Ba 
7 | .093037 Oe 
8 094047 16.82 
9 |. 095056 | 48"rr 
10 | .096062 16. “9 
11 | 9.097065 es 
2) 098066 | ane 
13 | .099065 16 62 
14 | .100062 | j¢°ee 
15 | .101056 | 16 53 
16 | .102048 16.48 
17 3308087 | “Fea 
18 | .104025 16. Hs 
19 ||; *.105010 | “Fe *5e 
590¢ =< 
20 | 105092 | 36°35 
21 | 9.106973 | 16 5 
22 | .107951 | i¢°5, 
23 | .108927 16 23 
24 | 3109901 | As 
25 | 110873 | 16-20 
26 | .t11842 | 18-18 
27 | 112809 | 46°08 
28 | 118774 | 46'o 
29) .114737 16 02 
30 .115698 15.97 
31 | 9.116656 
BONO | aa 
33 | (118567 veg 
34 | .119519 is 
35 | 120469 | 15.88 
86 | (121417 Hee 
7 | .122362 15 ae 
38 |  .123306 | 15 
89 | .124248 | 465 
41 | 9.126125 
42 | .127060 Hes 
43 )x-doggo8 | (20x82 
44 | 198925 | 18-58 
45 | 129854 oo 
46 | .130781 15 49 
7 | .181706 | 33°40 
48 | .132630 
- pas 15.35 
49 . 1383551 15.32 | 
50 | .134470 | 35°53 
51 | 9.135387 > 
p2 | .136303 | 15-27 
53 | .187216 | 42°65 
B4 | 188128 | “42°F 
55 | .139037 15 40 
B6 | 189944 | 45°49 
7 | .140850 | 32"o» 
58 | .141754 nop 
59 | .142655 we 
60 | 9.148555 | °°: 
/ | Cosine. | D, 1” 


Dy 1s | Cosine. 


9.996751 
. 996735 
. 996720 
. 996704 
. 996688 
. 996673 
. 996657 
. 996641 
. 996625 
. 996610 
. 996594 


. 996578 

. 996562 
. 996546 
. 996530 
. 996514 
. 996498 


ive} 


996482 


. 996465 
. 996449 
. 996433 


9.996417 


. 896400 
.996884 


.§96368 


. 996351 


996335 


. 996318 


-986302 | 


$9285 
996269 
996252 
996235 
996219 
. 996202 
996185 
996168 
996151 
996134 
996117 
996100 
996083 
996066 
996049 
996032 
996015 
995998 
995980 
995963 
995946 
995928 


9.995911 


.995894. | 


995876 
. 995859 
. 995841 
995823 
. 995806 
. 995788 
99577 


Sine. 


D. 1". || Tang. Doi. | Cotang. ? 
a7 || 9-089144 | 4» 99 | 10.910856 | 60 
95 || -090187 | jn er 909813 | 59 
o7 091228 | 4729 .§08%72-} 58 
OF 092266 | 305m GUT%84 | 57 
“On 092802 | yn os .§06698 | 56 
On 094586 | 4°59 .90E664 | 55 
"ov || -095867 | jaya |  .804638 | 54 
“Sy || 096895 | 1712 902605 | 53 
$25 097422 | jn’ go 802578 | 52 
“Or 098446 | yo" p9 901554 | 51 
OW .099468 16.98 . 900532 | 50 
oy || 9.100487 | 46 os | 10.899513 | 49 
OW 101504 | 3695 .§98496 | 48 
joe .102519 | 3996 697481 | 47 
“On 108582 | jg'g3 | 896468 | 46 
oy || 204542 | 679 | -£95458 | 45 
pas -105550 | 46° re 694450 | 44 
“ae -106556 | 3¢ "ro 893444 | 43 
97 || -107559 | 36" 68 .€92441 | 42 
On | .108560 | 36 '¢5 891440 | 41 
a | ORAC . 48 
‘oy || shea 16 62 .890441 |} 40 

|| 9.110556 ro | 10.889444 | 39 
28 || aat551 | 18-58 | eeeaao | 38 
Sa 112543 | 3659 | 887457 | 87 
| “og -113583 | 36° gr .886467 | 36 
ae 114521 16 43 885479 | 35 
“38 -115507 | 36°49 884498 | 2 
| oe -116491 | 3¢"95 .€83509 | 33 
98 117472 | 4633 .€82528 | 82 
oy || -218452 | 36°58 .881548 | 31 
So «|| («119429 16.25 880571 |; 80 
og || 9-120404 | 46 a | 10.879596 | 29 
‘or || 121877 | 648 878623 | 28 
“9g || -122848 | 4645 877652 | 27 
og || 128817 |. 36745 876683 | 26 
| ‘og || -124284 | 368 '9g 875716 | 25 
og || -125249 | 36°03 874751 | 24 
oR :126211-| 36°90 873789 | 23 
"og || 12792 | 55" ge .872828 | 22 
9g ||, -128130 15 95 .871870 | 21 

2 &O | € lo st FOOTE € 
3 scam | | eae 
28 || 1ag0coa | 19-88 |“ “eego06 | 18 
“38 131944 | 32" g0 868056 | 17 
“58 182893 | 32m .867107 | 16 
| ‘og || 188889 | 92° p5 866161 | 15 
"30 (|| +184%84 | 45 '29 865216 | 14 
‘og || +185726 | 55 '68 864274 | 13 
38 126667 | 35" ¢3 .863333 | 12 
“30 187605 | 45°69 .862395 | 11 
"og || -188542 | 55's .861458 | 10 
9.189476 10.860524 | 9 
88 || 1240409 | 18-85 |" “esoso1 | 8 
"98 -141840 | 35°48 .858660 | 7 
“30 142269 | 35°45 857731 | 6 
30 148196 | 45°49 .256804 | 5 
"OR 144121 | 35°99 855879 | 4 
“30 145044 | 45°37 854956 | 3 
“38 145966 15 32 854034 | 2 
30 146885 | 45°39 858115 | 1 
os 9.147803 | | 10.852197 | 0 
Deg Cotang. | D. 1’ Tang. 


Sine. 


| 9.143555 
144453 
. 145349 
. 146243 
147136 
. 148026 
148915 
. 149802 
. 150686 
. 151569 
152451 


9.153330 
154208 
155083 
155957 
. 156830 
157700 
158569 
159435 
160301 
.161164 


9.162025 
162885 
163748 
164600 
. 165454 
. 166307 
167159 
. 168008 
168856 
169702 

9.170547 
171389 
172230 
173070 
173903 
174744 
175578 
176411 
1V7 ate 


.178072 


| 9.178900 
2179726 
. 180551 
.181374 
. 182196 
. 183016 
. 183834 
.184651 

185466 | 
| . 186280 
| 9.187092 
| .187903 
.188712 
» 189519 

. 190825 
.191180 

.191933 
.192734 
* 193584 
9.194332 


D2". | Cosine. 


COSINES, TANGENTS, AND COTANGENTS. 


Tang. 


| 9. 995753 
995735 | 
995717 
. 995699 | 
. 995681 
. 995664 
. 995646 
b 99; 5628 
. 995610 
. 995591 
. 995573 


995555 
. 995537 
- 995519 
995501 
995482 
995464 
. 995446 
- 995427 
995409 
. 995390 


| 9. 995372 
995353 
. 995334 
. 995316 
. 995297 
.995278 
. 995260 
995241 
995222 


995203 


995184 
. 995165 
. 995146 
995127 
.995108 
.995089 
995070 
.995051 
. 995032 
. 995013 


9. 994995 
994974 
994955 
994935 
. 994926 
.994856 
994877 
. 994857 
.994838 
.994818 


994798 
99477 
994759 
994739 
994720 
994700 
994680 
994660 
| .994640 

9.994620 


io) 


| 9.147803 
|| (148718 
- 149682 
.150544 
.151454 
. 152363 
.158269 
154174 
.155077 
.155978 
. 156877 


1 wre 


Cbbe 


158671 
. 159565 
160457 
161347 
162286 
163128 
. 164008 
.164892 
165774 | 
| 9.166654 


167582 


. 168409 
. 169284. 
.170157 
.171029 
.171899 

172767 
. 178634 
. 174499 
.175362 
. 176224 
177084. 
.177942 
.178799 
.179655 
. 180508 
. 181360 
. 182211 
. 183059 


.188907 
.18475 52 
185597 

. 186439 
187280 

.188120 
. 188958 
.189794 
. 190629 
. 191462 | 


192294 
193124 
.193953 
.194780 
. 195606 
.196430 
.197253 
198074 
. 198894 
9.199713 


Cotang. 


| 10.852197 


851282 
850368 
849456 
848546 
847637 
846731 
845826 
844923 
844022 | 
843123 

10. 842225 
841329 
(840435 
839543 
838653 
837764 
836877 
835992 
835108 
834226 


10.833346 
832468 
831591 
830716 
829843 
828971 
828101 | 
827233 
826366 
825501 

10. 824638 
823776 | 
822916 
822058 


.821201 | 2 


820345 
819492 | 
818640 
817789 
816941 
10.816098 
815248 
814403 
813561 
812720 
811880 
811042 
810206 
809371 
- 808538 


| -10.807706 


806876 

806047 
. 805220 
. 804894 
~ 808570 
802747 
"801926 
.801106 


10.800287 


Cosine. 


| 
|} 


Sine. 


Cotang. 


Tang. 


fone whe sie 


Or DO CoP OF 


~ 


eS 
CODNOTR WWE OS | 


Sine. 


9.194832 
195129 
1195925 
.196719 
197511 
. 198802 
.199091 
.199879 
200666 
.201451 


(202234 | 


9.203017 
253797 
204577 
205854 
-206131 
. 206906 
207679 


208452 


209222 


. 209992 


210760 
.211526 
.212291 
.218055 
213818 
214579 
.215838 
216097 
216854 
217609 
9.218363 
.219116 
. 219868 
220618 
221867 
.222115 
222861 


2238606 


i=) 


224349 | 


225092 
9.225833 
226573 
220311 
228048 
228784 


2229518 | 


280252 
230984 
231715 
232444 
.233172 
. 233899 
234625 
.235349 
.236073 


ve) 


.286795 


237515 


238235 


238958 


9.239670 


o . . . . . . . 


Jes} 


TABLE XXV.—LOGARITHMIC SINKS, 


Tang. 


9.199713 
200529 
201345 

| , 202159 
202971 
203782 
204592 
205400 
206207 
207013 
207817 

9.208619 

| .209420 

210220 

211018 

211815 

212611 

213405 

214198 

214989 

| 215780 

9.216568 
217356 
218142 
218926 
219710 
220492 
221272 
222052 
222830 
223607 

| 9224382 

225156 

225929 

226700 

227471 

228239 

. 229007 

22977 

. 230539 

.231302 


9.282065 
232826 
233586 
234345 
235108 
235859 
2086614 
237368 
2388120 
238872 

| 9.289622 

| .240871 

241118 


248354 
. 244097 
244839 
245579 
9.246319 


241865 
242610 


fre re pe fred fh fee fed peek eek 


WWW WWWWWHO 


Cosine. 


Cotang. 


o 


| 10.800287 
| 


799471 
T9BG55 
797841 
797029 
796218 
795408 
794600 
793793 
792987 | 
792183 

10.791381 
"790580 
“789780 
788982 
“788185 
787389 
786595 
785802 
"785011 
784220 


10.7834382 


10. 


10.767935 | 


. 768386 
7162652 
61550 
761128 | 


10.760378 


158629 
758882 
758185 
757890 
756646 
755908 
755161 
154421 
10.753681 


Tang. 


Sine. 


Cosine. 


DrIDvwk WOH OS 


=) 


ao) 


. 239670 


. 240386 
.241101 


.241814 


242526 


. 243237 
-243947 
244656 
. 245363 
. 246069 
246775 
247478 
248181 
. 248883 


249583 
250282 
250980 | 


201677 
.252373 
.252067 
208761 
254453 


255144 | 


250884 
- 200523 
ania 
201898 
208583 
259268 
.209951 
.260633 
. 261314 
.261994 
262673 


.263351 


264027 
264708 
265377 


266051 
. 266723 


207395 


268065 


. 268734 
. 269402 
. 270069 


.270735 
.271400 
.272064 
212126 


. 273888 
. 274049 
.274708 
275867 


276025 


276681 | 


277339 


wh 


277991 | 
278645 | 


219297 


. 279948 


9.280599 


DNF AP ay 7 


=) 


Oo 


Ve) 


Je) 


9.993351 
993329 
993307 
993284 
993262 
993240 
993217 
993195 
993172 
993149 | 
993127 
993104 
993081 
993059 
993036 
992013 
|| .992990 
= || 992067 
992944 
992921 
992898 


. 992875 
. 992852 
992829 
. 992806 
', 992783 
. 992759 
992736 
992713 
- 992690 

. 992666 


992648 
. 992619 
. 992596 
992572 
. 992549 
. 992525 
. 992501 
. 992478 
. 992454 
. 992430 
992406 
. 992382 

. 992859 
- 992335 
992311 | 
- 992287 
. 992268 
992239 
. 992214 
. 992190 


992166 
992142 
992118 
.992093. | 
992069 
992044 
992020 
991996 | 
~ || .991971 

|| 9.991947 


| Cosine. | D. 1". | 


Sine. 


COSINES, TANGENTS, AND COTANGENTS, 


Tang. 


9.246319 
.RAT057 
-RATT94 
. 248530 
249264 
. 249998 
. 250730 
.251461 


.252191 


252920 
203648 
9.254374 


.255100 
255824 | 


206547 
257269 
.297990 
208710 
259429 
. 260146 
. 260868 
9.261578 
202292 
. 268005 
268717 


. 264428 


.265138 
. 265847 | 


. 266555 
.267261 
.267967 
9.268671 
. 269375 
270077 
210079 
201479 
202178 


212876 | 


27857 
244269 
274964 


9.275658 
. 276351 
.277043 
277784 
. 278424 
.279113 
.279801 
280488 
281174 
.281858 

9 282542 
. 283225 
.283907 
. 284588 
285268 
. 285947 
286624 
.287301 
287977 

9.288652 


Cotang. 


10.%53681 

152943 
752206 

751470 | 2 
750786 

. 750002 

.T49270 

748539 

747809 
747080 
746852 
745626 

. 744900 
T4416 

7438453 
742731 | 
742010 
741230 
740571 
. (89854 
739137 
«738422 
737708 

786995 
736283 
135572 
. 134862 
(84153 
. 188445 
132739 
732033 
731329 
730625 
(29923 
(29221 


728521 


nr 
1271822 


(a 


727194 
726427 
725731 
725036 
724842 
7123649 
722957 
- 122266 
21576 
20887 
720199 
19512 
T1886 

718142 

717458 

T1675 


(40 


- 716093 


712699 
712023 
| 10.711348 


| Cotang. 


Tang. 


CHW W ROOM =300 


TABLH AXV.—LOGARITHMIC SINES, 


WOMOIAIP WWE © 


ile) 


=) 


We) 


Sine. 


Cosine. 


9.280599 


281248 
281897 
282544 
.283190 
283836 


284480 


285124 
285766 
286408 
287048 
287688 
288326 
288964 
.289600 
290236 
. 290870 
291504 
.292137 
292768 
293399 
294029 
294658 
295286 
.295913 
.296539 
297164 


297788 


.298412 
299034 
.299655 
ae 
. 80089! 

"301514 
802182 
802748 
803364 
808979 
.804593 
805207 
.805819 
806430 
807041 
.807650 


808259 | 
. 808867 


. 809474 
.810080 


.3 10685 | 


.811289 
f° 811893 


812495 
.313097 
313698 


14297 | 


.314897 
.815495 

.816092 
.3816689 
2317 ae 


9.31787 


Cosine. 


ive) 


v=) 


9.991947 
991922 
.991897 
.991873 
.991848 
.991823 
.991799 
99177 
.991749 
99172 
.991699 


| 9.991674 
.991649 
. 991624 
.991599 
991574 
. 991549 
. 991524 


991498 
991473 
991448 


| 9 991422 


.9913897 
991372 
. 991346 
. 991321 
.991295 
.991270 


.991244 


,991218 
.9911938 
991167 
.991141 


. 991115 


. 991090 
. 991064 
. 991088 


.991012 
. 990986 
. 990960 
. 990934 


.990908 
. 990882 
. 990855 
. 990829 
. 990803 | 
990777 | 
. 990750 
. 990724 
. 990697 
. 990671 
990645 
. 990618 
. 990591 
990565. | 
990538 | 
. 990511 
. 990485 
. 990458 
990431 
9.990404 


Tang. 


289826 
289999 
290671 
. 291342 
. 292013 
292682 
. 293350 
.294017 
294684 
.295349 


296013 
296677 
297389 
298001 
. 298662 
299322 
.299980 
. 800638 
801295 
.801951 


802607 
.808261 
.803914 
304567 
.805218 
.805869 
. 806519 
.807168 
.807816 
.808463 
.809109 
809754 
.810399 
.811042 
.811685 
-812827 
.812968 
.813608 
-3014247 
.814885 


815523 
.3816159 
. 816795 
817480 
.818064 
318697 
819330 


xo 


ie) 


Je) 


© 


.819961 


«820592 


821222 


9.821851 


322479 | 


-823106 
.823783 
B24 358 

824983 
"305607 
326231 
.826853 


|| 9.827475 


| 9.288652 


Cotang. 


08 3393 


y 0p 661 
.701999 
. 701838 
.700678 
700020 
. 699362 
.698705 
.698049 
.697393 
. 6967389 
.696086 
. 695433 
.694782 
,694131 
.693481 
. 692832 
.692184 
.6915387 
| 10.690891 
690246 
. 689601 
.688958 


688315 


687673 
. 687032 
686392 
.685753 
. 685115 
684477 
-683841 
683205 
682570 
.681936 
.681303 
.680670 


680039 


679408 
678778 
| 10.678149 | 
.677521 
676894 | 
676267 | 
675642 | 
.675017 
.674893 
.673769 
673147 
10.672525 


Sine. 


Cotang. | 


Tang. 


10.711348 
- 710674 
-710001 
- 709829 
708658 
TOV 987 
707318 
106650 
705983 

705316 

704651 
: f (08987 ( 


COSINES, TANGENTS, AND COTANGENTS. 


9.317879 


o 
—) 


- 


= 
SO OID OS CO 2 


i=) 


=) 


Sine. 


Cosine. 


Tang. 


Cotang. 


.318473 
.819066 
.3819658 
820249 
.820840 
.821480 
.822019 


‘ .322607 


823194 


323780 | 


. 824366 
. 824950 
.825534 
.826117 
.828700 
827281 
.3827862 
20442 
.829021 
-829599 


.330176 


3380753 
.831329 
.331903 


.832478 


.833051 | 


.333624 
~334195 
.834767 


835337 


.835906 
.836475 
.887048 
.337610 
.838176 


888742. | 
. 889307 | 
.839871 


.3840484 
. 840996 
841558 
842119 
842679 


.843239 | 


843797 
844355 


.844912 


845469 


.846024 | 


346579 
847134 
347687 
348240 
.348792 


3349343 | 


. 849893 
.8504438 
.850992 
.851540 


| 9.352088 


We} 


Ve) 


eo) 


| 9.990404 
990378 | 
990351 
. 990824 

990297 
.990270 
. 990243 
.990215 
.990188 

.990161 
. 990134 


9.990107 
.990079 
“990052 
990025 
989997 
989970 
989942 
989915 
989887 
989860 
989832 
. 989804 
989777 
989749 
989721 
989693 
989665 
989637 
989610 
989582 | 


.989553 
. 989525 
.989497 
. 989469 
. 989441 
.989413 
. 989385 
. 989356 
. 989328 
. 989300 


989271 
. 989243 
. 989214 
989186 

. 989157 

. 989128 
989100 
989071 
. 989042 
. 989014 
. 988985 
988956 
988927 
988898 
988869 

. 988840 
98881 1 
988782 
988753 

. 988724 


io!) 


Je) 


ile) 


|| 9.827445 


o28095 
Oso 1d 
O2Id384 
.8 29953 
.830570 
.081187 
.831803 
.082418 
.0830383 
.883646 
804259 
.384871 
. 800482 
.8386093 
886702 
837311 
.837919 
.o80027 
.3889133 
.8097389 
.840344 
.840948 
-041552 
842155 
042757 
.343358 
.343958 
.844558 
-340157 
. 845755 
. 846353 
346949 
.047545 


(Ten) 
len) 


848141 | 


348735 
349829 
349922 
350514 
351106 
351697 
352287 
352876 
353465 
354053 
.854640 
355227 
855813 
. 3806398 
356982 
357566 
358149 
858731 
.009313 
359893 
360474 
361053 
361632 
362210 
362787 


i=) 


|| 9.363364 


a vm 


PJ FFF IIS 


| 10.672525 
671905 
671285 
670666 
670047 
669430 
668813 
668197 
667582 
666967 
666354 

10. 665741 
665129 
664518 
663907 
663298 
662689 
662681 
661473 
660867 
660261 | 

10.659656 
659052 | § 
658448 | : 
657845 


.658647 | 
.658051 
.652455 | 
.651859 
. 651265 
. 650671 
. 650078 
.649486 
. 648894. 
. 648303 
| 10.647713 
.647124 
. 646535 | 
. 645947 
. 645360 
-644773 
.644187 
. 648602 
.648018 
. 642434 


.641851 

.641269 
.640687 
.640107 
. 6389526 
.6388947 
. 688368 


.6387790 


637213 
636636 


e NaCosimes. |.) 1. 


Sine, 


Cotang. 


Tang. | 


TABLE XXV.—LOGARITHMIC SINES, 


- Sine. D. 1". |} Cosine. | D. 1’. Tang. D. 1°. | Cotang. / 
0 | 9.352088 | 4 45 |) 9.989724 4g || 9-363364 | 9 ,, | 10.636636 | 60 

1) 352685 | 9°79 |, 988695 | -48 ||” “363940 958 |. -636060 | 59 

2| 353181 | 9: 988666; 28 || “364515 635485 | 58 

3 | 1353726 , a 988636.) a | 1365090 : » 634910 | 57 

4] .354271 oi . 988607 0 |. 865664 “Of | 684836 | 5G 

5 | 1354815 | 9-07 .988578 | “25 || "366937 | 9-55 | “gagren | me 

6 | 355358 | 9-05 988548 | -0 .366810 | 9-53 | “gssig9 | 54 

7 | 1355901 | 9-95 ‘gsss19 | -48 -367382 | 9-53 |  “gg0618 | 53 

8 | .366443 | 9-03 || ‘osgigg BO || 1867958 oe | 682047 | 52 

9 | .356984 | 3-02 988460 | -48 368524 | 9-7. | 631476 | 51 

10 | .857524 9/00 988430 “48 || .369094 9.48 .680906 | 50 

| 11 / 9.358064 | ¢ og || 9.988401} |", || 9.869663 | 9.4 | 10.630837 | 49 
Hh 12 | 358603 | 8-88 988371 | .*:50 870282 | 9-42 | 1629768 | 48 
Hn 13 | 5941 | 8.97 988342} «38 370799 | 3-42 | 629201 | 47 
14 | .359678 8195 .988312 | , 50 871367 : 3 . 628633 | 46 

15 | .360215 | ‘8.95 988282 | © -> .371933 | 9-4 628067 | 45 

HW 16 | 360752 Sieg 988252 | 80 || “372499 3-42 | 627501 | 44 
7 |. .Beien7 | 8.82 -988223°| «48 373064 | 9-4 | “620936 | 43 

uh 18 | 361822 | 8.92 988193 | -P? 873629 | 9°45 | .626871 | 42 
Hi | 19 | 362856 | 8.90 988163} -P0 374193 | 3-38 | 625807 | 41 
He 20 | 362889 | 8-88 988133 | - 0 874756 | 8°38 | 625244 | 40 
AV ig 21 | 9.363422 | 2°. || 9.988103 50 || 9-875819 | 9 » | 10.624681 | 39 
vil ,22 | 363954 | 8.87 -988073'| +0 875881 | 9°34 | 624119 | 38 
23 | .364485 885 . 988043 50 3876442 9.35 623558 | 87 
24 | 365016 | 8-8 988013 | <P? 3770038 | 9°39 | 622997 | 36 

25 | 365546 | 8.83 987983} :P0 377563 | 9°35 | 1622437 | 35 

26 | 266075 | 8-82 987953. | 80 878122 | 8°35 | 621878 | 34 

27 | 366604 | 8-89 -op7ge2 | <b -378681 | 9°55 | 621319 | 33 

28 | 367131 | 8.78 987892. | -p0 379289 | 9°39 | .620761 | 32 

Hk 29 | 367659 | 8-80 987862 | 20 879797 | 9°58 | 1620208 | 31 
il 30 | 868185 | 8.7 987832 | <7 380354 | 9°59 | 1619646 | 30 
A 81 | 9.368711 | gos || 9.987801 | ~',) |) 9.s80910 9.97 | 10.619090 | 29 
ol 32 | .869236 ue 987771 "Po «|| «6881466 ee .618534 | 28 
i 33 | 369761 | 8-75 -987740 | >? || 1389000 | 9-23 | “gizago | oy 
Hil B4| 870285 | 8-2 || ‘osrrto PO |) 1882575 O33 | 1617425 | 26 
85 | .870808 | §.72 987679 | -B 883129 | 6°53 | 1616871 | 95 

36 | 371330 | 8-7 987649 | -20 "383682 “616318 | 24 

37 | 371852 | 8.70 "987618 |  -52 884234 | 9-20 | “i566 | og 
38 | 1372373 oon 987588 | «0 "384786 ae "615214 | 29 

39 | 372804 | 8.68 987557 | Fp || -ae5aaz | 8-18] “614663 | 21 

AQ, : AOR .t KC . ¢ 

40] .avsai4 | 8-67 987526 | Fo || 885888 | 9-18) 614112 | 20 

41 | 9.373933 | .'g. || 9.987496 se || 9386438 | 9 4. | 10.618562 | 19 

42 | 374452 863 987465 =) .886987 9°15 613013 | 18 
43 | .374970 8 62 987434 “B0 887536 9°13 .612464 | 17 

el weeeeee | geo || e8C40B: | “2 -888084 | g'i9 | 611916 | 16 

45 | .376003 | §-60 ‘perei2 |. 62 888631 | 9°15 | 1611869 | 15 

46 | .876519 8 60 987341 “BO .889178 9°10 .610822 | 14 

7| .3770a5 | 8.60 987310 | - 389724 | 9°49 |  .610276 | 13 

48] ‘8rrsz9 | S87 987279 ae | "390270 a 609730 | 12 

49 | .378063 g RY . 987248 “30 890815 908 609185 | 11 

50} .s7s577 | 8.5% 987217 | “Pe 391360 | gps | .608640 | 10 

51 | 9°379089 | . +. | 9.987186 > _ || 9.891903 ~ | 10.608097 | 9 

52) .879601 | pss || 987155 | -P2 || ‘390447 | 9:07 | “ “Gomnca | 

53} .880113 | §-°8 | ‘ogvioa | -52 || “3959g9 9.03: “go70i1 | 7 

54] (880624 | 8-52 || “oproga | — -58 393531 | 2-93 |  “eog469 | 6 

55 | .38iis4 | 8-90 || “oszoei | 52 |! “304073 303) 1605927 | 5 

56 | .881643 | gv4s || -9svo30 | -82 |) “Baaeia 900 |  -605886 | 4 

b7 | 882152 | ee || .986998 as , 395154 9°00 604846 | 3 

58) .382661 | oye || .986967 “25 «|| «| «6895694 8/98 604306 | 2 

59 | 883168 | B42 || “986936 22 396283 | Bor | .603767 | 1 

60 | 9.383675 | °-4° || g gseg04 | 58 |) g Bggen °-3¢ | 10603229 | 0 

’ | Cosine. | D.1”. || Sine. D. 1". 4| Cotang: | D. 1°. Tang. 2 


- 


Sine. 


SO OI OTR OO DH OS 


9.388675 
.304182 
.884687 
.885192 
.385697 

| - 886201 

.886704 

.881207 

.887709 

.3888210 

.o88711 


9.389211 
889711 
.890210 
.390708 
.891206 
.3891703 
.3892199 
.3892695 


.893191 | 


.393685 
| 9.394179 
| .394673 
395166 
895658 
396150 
396644 
897132 
897621 


898111 


.3898600 
9.399088 


899575 
.400062 


.400549 
.401035 
.401520 
.402005 
.402489 
.402972 
403455 
9.403938 
.404420 
.404901 
.405382 
.405862 
.406341 
.406820 


407299 


40777 
408254 
9.408731 
.409207 
.409682 
.410157 
.410632 
.411106 
.411579 
.412052 
412524 
9.412996 


| 


10 GD 00 CO OO 


CO 


wiowwwwwe a aye ee 


z| Cosine. ! D. 


COSINES, TANGENTS, AND COTANGENTS, 


Cosine. | D, 1". i Tang. Cotang. 
= es | 
9.986904 | 9.896771 | 2 10603229 
. 986873 .397309 |g” 602691 
:. 986841 897846 | 9° 602154 
986809 898883 |g" 601617 
1.986778 |} .898919 |g" 60081 
-986746 || 899455 | 9° 600545 
986714 |} 899990 | 9" .600010 
. 986683 |} -400524 | ¢" 599476 
(986651 |} .401058 | 9°¢ 598942 
.986619 || .401591 | 9’ 598409 
. 986587 || -402124 | 9° 597876 
| 9.986555 || 9.402656 | 10.597344 
986523 || .408187 | 9° .596813 
. 986491 |} 408718 | Qos 596282 
. 986459 || .404249 | 9° 595751 
-. 986427 || 404778 | 9’. 595222 
986395 |} -405808 | g'¢ 594692 
:986363 || .405836 } 9'¢ 594164 
986331 .406364 | 9° .593636 
986299 || .406892 | ¢° 593108 
986266 || .407419 | 9’ 592581 
9.986234 | 9.407945 |g mm | 10.592055 
986202 || -408471 | g'ns 591529 
.986169 .408996 | g'nx .591004 
.986137 409521 | g"hg 590479 
. 986104 .410045 | 9'r3 589955 
. 986072 | .410569 | Q"r5 589431 
.986039 | .411092 | p'n5 588908 
.986007 || .411615 | 9° 588385 
. 985974 .412137 | 9'¢ 587863 
,985942 || 412658 | 9" 587342 
9.985909 || 9.413179 10.526821 
985876 .413699 586301 
985843 .414219 | 9 585781 
985811 .414738 : 585262 
.985778 BB .415257 | 0’ ¢ 584743 
:985745 fee 415775 | 9" Ge 584225 
985712 | ‘ee 416293 | 9’ g¢ 583707 
985679 Bee .416810 | o'¢ 583190 
.985646 “3B 417826 | 9° 582074 
| 1985613 “BB A1T842 |g” 582158 
9. 985580 9.418358 | 10.581642 
ges5a7 | -P5 || 4ise73 | 3° 581127 
985514 os 419387 | 580613 | 17 
. 985480 5S .419901 -580099 | 16 
985447 55 420415 -519585 15 
985414 Bs || | -420927-| 919073 | 14 
985381 "no || 421440 578560 |, 13 
. 985314 “my || 422468 5775: (| il 
. 985280 — || 422074 577026 | 10 
9.985247 | nn || 9.423484 | 10.576516 | 9 
985213 “a5 || 428998 576007 | 8 
.985180 By ae oe 424503 5 4 5494 ¢ 
985146 "rs || -425011 | 574989 | 6 
985113 | “py || .425519 | 574481 | 5 
985079 | “py || .426027 573973 | 4 
985045 “ny || 420584 | 573466 | 8 
985011 “a5 || «427041 572959 | 2 
| .984978 oe || | Saeed? 572453 | 1 
|) 9.984944 | °°" |) 9.428052 10.571948 | 0 
Sine. . || Cotang. Tang. / 


& 


TA y 
\BLE XXV.—LOGARITHMIC SINES 


/ . 
Si " | 
ne, D. rt 4 | Cosine. D. 1’. Park : 
tk 3 Oe i | g. D. 1". | Cotang. vey. | 
0 | 9.412996 | 
1} 413467 | ig Oiuee (i Me: {eee | Lb | 
2 7.85 5 ¢ Py eae te ae i As | 
3 Siri 783 “981876 Br. |i ARSE ors ee | 6 | 
asiava | | 7488 984845 57 .429062: | 9° enn trie 
5 | <l4tead7 782 speaoa |) Bll gaunne oe “br0as4 | pe 
Pel Ghagseis | TSO mean, ‘57 || -420070 | 3°38 “560980 | e6 
7 | 416983 | 7.80 984740 | +e 430573 | 8-38 ga rae 
ol sttginey | TURD AH) eeeaten py || -431078 | B37 "B6R0D5 | bad 
10 | 417684. | 7:78 "984638 | -3¢ 432079 | 8-30 ERS. | oe 
| --417684 | ote 984688 | i5g || | - 482580 8.35 | -567921 | 52 | 
11| saisiso| || 9. ‘py || «483080 | 8-38 ‘GOtA2) | Od 
te | 418615 | 0:5 ee ma ie sanege 8.33 .566920 | 50 
3 re ” - 984535 teks : 
| wro | 28 |) $880) ) 38 |) es) sm 05000 | 48 
ig \ aseoor |) 782° deeaaee | 434579 | 8-82 | “565401 | 47 
16 sas yh . 984432 157 .435078 32 Spec f 
17 | 1420983 ie 984397 58 ee 828 “56LRt | 45 
18 94205 Vaid . 984363 : bec vs : Or 
18 | 421395 | fy || 984828 |» Ft aosi0 | 828 | ‘563180 | 43 
oy tuessis (7 MOC Te Geers 4 Sette’, po seenden a 
ry RF As i 40 (ODE . F. ra ae . 
21 | 9.422778 (.67 . 984259 “58 ‘438059 | 8-2¢ 562437 | 41 
oe "493038 | 7.67 ae fire » || 9.438554 re aati 
93 92RQ7 ” 65 . 90 mY ae 55 ¢ ‘ : 
gy gases | ia osiibs | 3S 439048 | 835 ae Ie 
95 | “494615 | 7-65 . 984120 : ne 89s 56045" 
ised ) : 26 122 560457 Wg 
26 | .425073 | 1-68 984085 | Pe aoe tt Siae 559964 | 3¢ 
2 | 425580 | tgp |] 984015 | Ep ware ioe “ppe078 | 34 
oa) aogtes || estes (ic fee seen a0 | ree tae 
30 | capos09 | 7 epit||) woese4s |) Gee "442006 | 820 | “psraod | Sp 
426899 6 983911 .58 -442497 8.18 ; 55% 994 | 82 
ap | oiommad Ili grease “eo. || 3442988 | 8-38 557508 | 31 
32 | 427809 | 7-98 9.983875 ine 8.18 .557012 | 30 
82) 427809 | 757 || -983840 | G8 nei 10.5565 
gets | Lee ll| perce 8 Vaca Bo eee 
35 | .499170 | 7-55 9837 58 444458 | ooy 5 SeeSEta {7 
Fe) if en Vv 15 .55dDd42 or 
36 42962 7 5b . 9837385 .58 444947 RRRAre 5! 
pote tbe f “ganas | 8148%| eeteee | oe 
37 | 430075 | 708. || “ogenee eet atbi85 | lig | -554565 | 2 
38 | .4305e7 | 1-58 983664 .60 -445923 | oy: 554077 A 
39 | ‘430078 | 7-5 ieeseco |) 286. 1 caaeeae Age | Oe ee 
Bl eee | (rie 983620 | 53 || -440898 | 8°75 “553102 | 28 
a) Sr od : 2 6 ; > z . € ‘ pepe 
a1 | 9.4zis79 | gases | +82 || <aa7e70 | 8-10 552616 | 24 
. 432329 | 7.50 9. oe fo || 9.448858 8.10 .552130 | 20 
5 90nhh Lo A, wa RS Nya V0 y O50 
“4 | 433096 nay Spee || abe lt gates 7 LOG 
| dass, | TBO | pees “60 || 449826 | Boe pers al ae 
ae 3 ¢ 07 550674 | 17 
aa) Bias | eee nee 58 449810 | 8° gy Se reme 
47 As peng KY 5 983345 .60 450294 07 oO Le J 16 
| Bin | 18 | cies | 2 | ia | Bo igees | 1d 
49. |  A3b4g0 | 7-4 reece (200 tl eepneee 1 8 an ane 
40 | -aioaca | F-23 || cosas | BS || “apzoos 8.05 | -S4sos7 | Te 
) by Cag 2951)¢ : - 452225 Us -0F0e0¢4 | 12 
| 51 | 9.486353 fee re “60 “450706 | 8-02 547775 | 11 
52 | .436793 | 7.42 9.983166 | 9459187 8.02 .547294 | 10 
ba | 436798 | 7°79 || 988130 | “Gp Seen 5 | 10.5468 
54 | ‘43763 | 7-40 -83004 | “6° ee oe “BAG839 5 
55 | .438129 | 7.38 “ossosg | -89 |] “graeos | 8.00 BA5R5O | 
56 | 438572 | 7! (983022 | 80 454628 | 2-08 ee Le 
Br | 14a9014 (88 |} ‘oegs6 | — -60 “155586 098 ‘514808 | 
BY | 480014 | 757 |] -982950] “69 || 455586 | 7-38 |  paddi4 
59 139807 | 7-3 “98291 60 -456064 ‘. a7 082" 4 
60 | 91440585 | 7 35 speereeg ft) 200. Name eee Be: ieee |e 
Tait Wiccan ae pigeaeia (1.800 fl cee 7,05 5| | Soieee eg 
Te AEE : peer 9.457496 | @:9° .542981 | 1 
| Cosine. D. 1’ cee ap ae I 10.542504 0 
; real : oe 
otang. | D. 1 Tang. / 


=I 
TS 
oO 


COSINHS, TANGENTS, AND COTANGENTS. 


Cosine. | D. 1". |) Tang. D. 1”. | Cotang. | u 


Sine. Dales 


———— - —— | 


| 9.440388 | » 95 || 9.982842! | 9.457496 


; | 
10.542504 | 60 


| 0 Pe 

Pa) lagers | 4:83. || ““osesos | 62 || ““aszora | 5-88 542027 | 59 
2 | 441218 | 7:33 || Toseveo | 69. |) “assaag | 1:98 "541551 | 58 
Si} <4aress | 2:33 || ‘oseraa | 89° || “aseoan | 1-98 "541075 | 57 
Bat Serene | | Tepes | Berens |) BO |i cepa 7 ae PAGO) + O8 
5 | °,442535 3 . 982660 60 || .459875 90 ~b40125 . 55 
6 | .442973 og ||  - 982624 eat .460349 90 .5389651 | 54 
7| 443410 | 7°58 || -osess7 | -8 || ‘46ose3 | 1:00 ‘539177 | 53 
B | cft8847-| 4'5q! ||. oee5bt | Bo |) Vaeigor | 7280 "538703 52 
g| 444984] 2:25. || “gosta | 82 |) Cagi77 & "538230 51 
0 


444720 


11 | 9.445155 
12 | 445590 
13 | .446025 
14} .446459 
15.| 446893 
16| 447326 
17 | 447759 
18 | 448191 
19 | 448623 
20 | .449054 
449485 
22 | .449915 
23 | .450845 


te 537758 , 50 

10.537285 | 49 
85 "536814 | 48 
536342 | 
585872 | 46 
585401 | 45 
534931 
534461 | 43 
538992 | 42 
583523 | 41 
“80 .533055 | 40 
, 10.532587 | 39 
bd 531653 | 


— 


Os 982477 | 65 462242 
on || 9.982441 gg || 9.462715 
3: 982404 “69 .463186 
| asa | 8 || aes 
23 Nae 62 eine 
4 982294 a 62 : 464599 
39 982257 eg 465069 
30 982220 "62° || -465539 
50) . 982183 an .466008 
18 || -982146 Tay 466477 
18 .982109 "gg || -466945 
» || 9.982072 gg || 9-467413 
~ || 982085 .467880 
1” .981998 "69 468347 


aN 
~ 


iS 
ne 


ne) 
are 
ile) 


J 


WEAF AF QT APPIN AAI AHH NI WANN HAIN NAN 
2 é 
\ 


24] 1450775 | #75 .981961 | *¢5 .468814 ne .531186 | 36 
Q5 | 451204 | »'y5 /]| .981924) “63 .469280 wy 530720 | 35 
26 | .451632 13 981886 | “¢5 469746 5 530254 | 84 
27 | .452060 3 981849 | “65 ATO2A1 ne 529789 | 83 
28 . 452488 {27.| . 981812 63 .47(0676 a5 529824 82 
29 | .452915 19 98177 “ga || 4c114 73 528859 | 31 
30 |  .453342 10 981737 | G5 || 471605 °3 -528395 | 30 
31 | 9.453768 10 || 9-981700 gg || 9.472069 wo | 10.527931 | & 
82.| .454194 | »°p9 981662 | “65 472532 | nny 527468 | 28 
83.| 454619 08 981625 | “gg || .472005 | ne 527005 | 27 
4 | 455044 | 7'og |] 981587 | gg |] 48457 | aig | -BRODKB | 26 
85 | .455469 07 981549 go || -473919 Ff 526081 | 25 
86 | .455893 981512 | ‘eg || .474881 | »'¢ 525619 | 24 


525158 | 23 
.524697 | 22 
524237 | 21 


~ 
my, RoaKrin 20 


37 .456316 
38 .456739 
39 457162 


‘on || 981474 | 788 |] lazagae 
05. || -981486 | *g5 ||  .475308 
03 | ,981399 - 63 475763 


: 
> O> 
eM) 


> FFF HF BPD I VIII 
S 
Ww 


40 | 1457584 OeteEL |, Ses oll cateaes.| Oe08 | 1 ances 
| 41 | 9.458006 » || 9.981823 | gs || 9.476683 | » gs | 10.528317 | 19 
42.| 458427 | «*oo || .981285 | “ga || 477142 | 9 "¢e 522858 | 18 a 
43-| 1458848 | 4-72 || ‘ogioa7 | = 83 || larre01 3 | .592899 | 17 
44) 1459268 | 4-29 || ‘osi209 | -63 || -aze050 | f:a5 521941 | 16 
45 | 450688 | fy) || cosiizi | 68 || :4rssi7 | “fgg | 1521488 | 15 | 
46} .460108 | d-o3 || .981138| “83 || c4vsovs | #05 | ‘521025 | 14 
47 | .460527 | G93. || .981095 | “B38 | oS | 1520568 | 13 
48 | .460946 | 8-8" || ‘osios7 |  -B3 vanes : 12 
49 | 461364 | Poy || .981019| “es a 11 


“80 519199 | 10 


BU | .461182 | gos 1} .980981 
10.518743 | 


.462199 6 |} 9.980942 | 63. || 9.481257 


| of | 9 704 .95 Han 2 Ur 24449 58 290Q 
52 |  .462616 | 9°93 -980904 63 || 481712 58 518288 
| 538] .468032 | 698 . 980866 “65 482167 57 .517833 
54 18 | 698 .980827 ot 482621 5 514379 | 
Bb ant Owe 980789 ie 483075 ts 516925 | 
vo Qo . ( i. fr 
56 | Cane "980750 | ° 483529 of 516471 


oo 516018 | 
“8g 515565 


Bz | :4646oa | 8-22 || ‘ogovie | -B8 || -4gs982 
| 58 | ‘465108 | 9: 980673 | 63 || 484435 


Sc 
« ~ 
* 
ee 


aJ  a a NOM SS SS SON SS  S O S  O ed 


ORD WR OLD -IO0O 


| 50 | 1465522 | 8-9 || ‘ogosa5 | -68 || apager | 2:2: "515113 

| 60 | 9.465935 | ©-88 |) o’9s0596 | -® || 9.485839 | 4:3 | 10:514661 

| ‘| Cosine. | D.1”". || Sine. Dal" sai} Cotanes|\3Ds1" Tang. Z 
10° 720 


é Sine. | D.1". || Cosine. 
0 | 9.465935 | @ gg || 9.980596 
1 | .466348 | gga || 980558 
2 | .466761 | ¢'oo .980519 
3 | 467173 6 87 980480 
4 | .467585 6 ' 35 980442 
5 | .467996 | gs 980403 
6 | .468407 | @'g3 .980364 
7 | .468817 | 6 ’39 980325 
8 | .469227 | g’ga 980286 
9} .469637 | 6'g5 980247 
10 | .470016 | @"g5 980208 | 
11 | 9.470155 | 4 ay 9.980169 
12] .470863 | @g9 .980130 
13 | .471271 | bgp 980091 
14 | .471679 | Gr 980052 
15 | .472086°)  ¢ bad -980012 
16 | .4724192 67 979973 
17 | .472898 697 979934 
18 | .473304 | @'te 979895 
19 | .473710 | g"hs 979855 
20 | (474115 6.73 979816 
21 | 9.474519 | og 9.97977 
2) .474923 | @° "3 979737 
23 | 475827 |g "9 979697 
24] .475730 | p'n5 .979658 
25 | .476133 | g'n5 .979618 
26 | .476536 6 70 979579 
297 | .476938 6.7 979539 
28 | .477340 | ¢ 68 979499 
29 | 477741 | 66g 979459 
30 | .478142 | @ 7 979420 
31 | 9.478542 | ¢ gn 9.979380 
32 | .478942 | ¢° 67 979340 
33 | 479342 | 6g 979300 
84 | 479741 |g 65 979260 
35 | .480140 6 65 979220 
36 | .480539 | ¢" 63 979180 
37 | 480937 6 62 979140 
38 | 481334 | 6 '¢5 979100 
39 | .481731 | 6 '¢5 979059 
40 | .482128 | 669 979019 
41 | 9.482525 | & 9.978979 
42 | 482021 | 2: 58 978939 
43 | .483316 6 60 978898 
44 | .488712 | re 978858 
45 | .484107 6 87 978817 
46 | .484501 6 57 97807 
47 | .484895 6 BY 978737 
48 | .485289 | pe 978696 
49 | .485682 6 bE 978655 
50 | .486075 6 53 978615 
51 | 9.486467 | ¢ 5x 9.978574 
52 | 486860 | p> 978533 
53 | .487251 | @ “5g . 978493 
54 | 487643 | 655 978452 
55 | .488034 | ¢54 978411 
56 | .488424 6 50 978370 
5Y | .488814 650 978329 
58 | .489204 | ¢ “48 978288 
59 | .489593 6 48 978247 
60 | 9.489982 om tL 9.9%3206 
4 Cosine. | D 1”. Sine. 


TABLE XXV.—LOGARITHMIC SINES, 


| 
DD a** Tang. D, 1". | Cotang. | fi 
nf nae Sh | Aes ee 
bs 9.485839 » 5g | 10.514661 | 60 
63 ‘ag5vo1 | 2-58 "514209 | 59 
65 | 48022 | pp | 518758 | 8 
63 | casriss | 750 | cSrpepr | 56 
. JD AQUOS f.o S ony xe 
65 || 4agoa3 | 7:50 | °-Biigne | 
65 || “gggag0 | 2-48 "511508 | 53 
ee (|| 488941 | 248° | “511059 | 52 
"65 489390 a 7 510610 | 51 
te ||| teeeoees |; bee "510162 | 50 
QO? Ww 
| me | ray | sso | a 
ee || -491d80 | 2-45 | “s08se0 | 47 
+ Ue On . 972 | “AR 
67 7 ||) satenek |! reat woes al We 
65 ‘4go519 | 7-43 "507481 | 44 
‘65 || "499965 | 1-48 507035 | 43 
65 || agaaio | 7-42 506590 | 42 
. 92Q6 : 50614 | 
a eee cea 
7 |, sae. solo ee Ae 
65 || 9-494743 | gq | 10.505257 | 39 
+ Oe 51 504s | 9 
ai | 2 | Pi | 
OS || .496073 | 2-88 | ‘503927 | 36 
a | 496515 | 03% | 508485 | 35 
. ie AQ mOkNh -t . € . 
|) a | Far | See | 
67 |) “4g7g41 | 7-37 "502159 | $9 
.67 “agora || Tee ieaieia teke 
8) BES) eB | Sek a 
£67 ||| PCE otis souhee 
tr Wt Seana || eee |e ee 
67 Ap x ec 7 39 0) C Oe sf 28 
“|| 50002 | 4-82 499958 | 27 
. 2 : am ie oF 
S67. |} Gengasy (eae aaeeaa bene 
67 || “501359 | 7-82 “498641 | 24 
67 | “sor707 | 7-30 | “aoapo3 | 93 
-67 || “502035 | 1-30 497765 | 22 
oy || poze | 1-28 | ‘ag7goe | 91 
“ey . 503109 98 .496891 | 20 
~ || 9.508546] » 4. | 10.496454 | 19 
S| Rs | 2 | ae | 
67 Meagaxa {0 Viet “hy doo pl ef 
| a) ie | aie 
67 BO5724 | 0-20 494276 | 14 
a “506159 | 1:29 "493841 | 13 
08 |) soroer | 223) jours | ti 
“68 507460 | 9°55 .492540 | 10 
Weer? y 6) nw 
‘68. |) OP etOs | ibe Boe A 
6% || “s0a759 | 7-22 491241 | 4 
“ba (| -500191 | £-22 | 7490609 | 6 
OC NOR. peas ary 
| gee | Bons 
ea. {| .310485 | 2-48 | lago515 | 8 
"fg || - 510016 | 7:18 |  ‘apoosa | 2 | 
. K1794R8 . Nevada | 
$68. |) gesaeeca| (THEY AB88DE | | 
9.51177 10. 488224 0 
D. 1". || Cotang. | D: 1°. Tang. i 


BOIS OR G0 


OD Y Se * G 


Sine. 


.490371 


.490759 


.491147 
.491535 


.491922 
492308 


.492695 
.493081 
.493466 
493851 


9.494236 
.494621 
-495005 
- 4953888 
495772 
.496154 
.496537 
.496919 
.497301 
. 497682 

| 9.498064 

|  .498444 

498825 

. 499204 

499584 

.499963 

.500342 

.500721 


501099 


501476 


| 9.501854 

502231 
502607 
502984 
“503360 
503735 
504110 
504485 
“504860 
505234 
505608 
505981 
506354 
506727 
.507099 
OVE 
507843 
508214 
BOSS85 
508956 


| 9.509326 
| 509696 
510065 
510434 
510803 


© 


.511172 | 


.511540 
.511907 


512275 


| 9.512642 


0 | 9.489982 | 


eo} 


S Sd G2 ¢ > S> S> > > 
ee ee ek ek eek ek Re A 
WO WO 0 GY CLOT OT OT z 


| 9.978206 


.978165 
978124 
.978083 
978042 
. 978001 
977959 


977918 
ITBTT 
977835 
877794 


ioleds4 
. 97 ( (52 


=) 


ITT 


. 977669 
977628 
977586 
977544 
977503 


977461 


977419 
77377 
9.977335 
977293 


IV7251 


977209 
977167 
977125 
977083 


977041 


. 976999 
. 976957 
976914 
976872 
976830 
976787 
976745 
. 976702 
. 976660 
976617 
. 976574 
.976E32 
. 976489 
. 976446 
. 976404 


io) 


.976361 


.976318 


Or 


976275 
. 976282 
. 976189 
- 976146 


976103 


|| 9.976060 


975974 
. 975930 
. 975887 
. 975844 

. 975800 
ON 5 S57 
dIlodiodd 


975714 


9.975670 


Cosine. | 


A a a a a a a a a a 8 a a 


WWNHHWNWWNWDN YWWWNWNWHWWSOD WOWwWowsd 


Tang. 


9.511776 
|| 512206 
512635 
513064 
| 513493 
| 513921 
| 1514349 
514777 
515204 
515631 
| 516057 


9.516484 
516910 
517335 
517761 
518186 
518610 
519034 
519458 
519882 
520305 

9.520728 
521151 
521573 
521995 
522417 
522838 
523259 
523680 

| 524100 

| 524520 


9.524940 
.5253859 
525778 
.526197 
.526615 
.527083 

| 527451 

.527868 
528285 
528702 

9.529119 
.529535 
.529951 
.580866 
530781 
.531196 
.531611 

| 582025 

.532439 


See 
| .582853 


| 9.533266 
.538679 
.534092 
584504 
.534916 
585828 
.535739 
.536150 
5386561 

9.586972 


Cotang. 


WE AE AE AEA EVE APE EEE PEI IE I IM N AIAN 


DAIMDMDADAA ADA SD 


oS 


APOIAS 


DS SH SG. Sd 


487794 
487365 
.486956 
.486507 
.486079 
.485651 
485223 
.484796 
.484369 
483943 


| 10.483516 


483090 
482665 
"482239 
481814 
481390 
480966 
480542 
480118 
479695 


10.479272 
478849 
478427 
-478005 
477583 
477162 
476741 
476820 
475900 
475480 


10.475060 
474641 
474222 
-473803 
473885 
472967 
472549 
472182 
41715 
.471298 


| 10.470881 


470465 
-470049 
.469634 
.469219 
468804 
-468389 
467975 
467561 
467147 


| 10.466734 


466321 
-465908 
.465496 
465084 
-464672 
.464261 
-463850 
.463439 
10468028 


Cosine, | 


Sine. 


Cotang. 


Tang. 


| 10.488224 


Or dD CO O10 FF OC 


~ 


~t | 
= | 
° 


TABLE XXV.—LOGARITHMIC SINES, 


~ 


Sine. D. 1", || Cosine. | D. 1’. || Tang. D. 1". | Cotang. | é 


i 
| 
1 


| 9.512642 


0 | > || 9.975670] ». || 9.586972 463028 | 60 | 
1| .513009 | ©-12 || “‘ovsgo7 |. -72 Roreae |) 08880 || VRE eed 
3 | ‘513375 | 6-10. || “Ooeess | 78 || “pares? | gigg, | | +462618)) 59 
| -513375 | g°49 975583 73 || -5B0792 | Bes .462208 | 58 | 
3 | 513741 | 6-1) 975539 mp || 388202 | 6-88 1461798 | 57 
4! .5iai07 | 8-18 975496 | “8 5as611 | 6-82 "461389 | 56 | 
5) 514472 |g og 975452 73 539020 | B95 460980 | 55 | 
6) 1514837 | Bp3 || 975408.) -23 suodeg | 6682 ‘460571 | 54: | 
al sper 6:07 =a “ns 539837 | go .460163 | 53 | 
.51556 at By . 975382 Seo. it rode oe 459755 | 52 
Peon | 6.0 ua ore E | 2 oO 409 (00 ve 
9 515930 | mae 915207 oh ‘| .540653 | oe "459347 | 51 
10 | 516204 | 6-07 975283 | +3 541061 | 6-8 “458939 | 50 
11 | 9.516657 | .,~ || 9.975189; “.. || g.saiaeg| 458532 | 49 | 
12 | 5170290 | 8-05 ‘ovis | 73 || OReere | 6:78 yrereee 48 
18 | 1517882 | 6-938 || “opsioy | -73 |] ‘54008] | 6.77 457719 | 47 | 
| 14 | 517745 | 8-0 |) lov5057 | 73 || ‘sageng | 6:78 | “asraie | 46 | 
i 15 | 1518107 | 6-03 || ‘975018 | 73 |) “543004 | 8-7 |. “azgooe | 45_| 
i 16 | .518468 | 8-92 || “g749g9 | 48 ‘543499 | 6.75 “4a56501 | 4a. | 
7 | .518829 | 6-9 || “oragos 13 |] “543905 | 6.77 456095 | 43 
| 18 | 1519190 | 8-08 ‘g7asgo | 45 544319 | 6.75 “455690 | 42 
i 19 | 1519551 | 8-02 974936 |  -73 544715 | 6.45 "455285 | 41 
i, 90] “teipo11:|, 82007 || “Bean ||, ca || CAPER | gong, || TUdbBeBE A Mt 
a 51S au 974792 bs 545119 | 6-23 -454881 | 40. | 
id 21 | 9.520271 9.974748 | ,. || 9.545504 | . |, 54476 | § 
Hn 22 | '520631 | 9:20 || “‘ozazo3 | -45 pépoee | eta) | eee | e 
| 23 | 520990 | 5-98 974659 | 43 546321 | 6.72 453669 | 37 
| 24 | ‘521349 | 5-98 |] “oraeia 05 "E4675 | 8-73 deriomertde Fe 
i 95 | _Bo1” Bu? || Oa and 13 marian | O.t2 pret E> 
Hi | uB@l707 |, Soa: || 97454 hi 547138 wis 452862 | 35 
| 26 | 522066 | 5-95 || “ovanes | -75 |] “Byenag | 6.70 452460 | 34 
Q7 | ‘5ee404 | 5.9% 974481 | +48 547943 | 6.72 “452057 | 3s 
28) 1522781 | 2-9 || ‘oraaag | «75 548345 | 8.70 “ae16e6 | 88 
29 | 1523138 | 5-95 || “ov4zo1 | -75 | sagyar | 6.70 | “gators | St 
fF 30] 523495. | 595 || 974347 |B || oaongg | 8-7 | asoas1 | 30 
| 31 | 9.523852 | 9.974302 ‘we || 9.54955 | 29 
Hi 32 | "1524208 | 5-93 || ““oyao57 | 75 || "pagern | 6.68 | ASOD | a8 
| 33 | 524564 | 2-28 974212 | ‘550352 | 6-68 449648 | 27 
| Fy ROAOs 5.9% Sea ay "5 ee iv tet kovihe Bs. 
Mt SA | £24920 | bop |] -9r4t0v | +f || “B50782 by | «449248 | 26 
i B | 2525271 me tT gvaig2 | «4 BB115: 68 | 47 
36 | 2525630 | 5-92 || ‘orgory | 73 || “perare | 6.65 prin 54 
Wt 37 | 1525984 | 5-90 974032 | +5 551952 | 8-67 "448048 | 23 
a, a8 | .526339 | 2-78 || covzos7 | -15 |) “aseas1 | 8-85 |  “agreag | 20 
an 39 | .526593 | 5.90 ‘973042 | «05 552750 | 6.65 447250 | 21 
Bilas | ornare 0.0 raoqr 45 ap Bon ; 5 - . 
a 40 | 527046 | 3°99. || .-978807 | -f2 |] 558149 | 6-65 | (446851 | 20 
iii 41 | 9.527400 : 9.973852 | *,, BBS 16452 
a 42 | “sav753 | 5-88 ovsco? | &73. || eeatan | oxeay | tOMaaRG ae 
Hi a5 | yes 5 97 | 91 380% ria -558946 6 63 .446054 | 18 
i a Lost! : 5 88 | aie “as 554344 6 62 .445656 | 17 
a 45 | ‘52esi0 | 8?) || ‘oreri | -75 || “pesigg | 6-68 | “Saagor | de 
ih 46 | ‘529161 | 9-85 973625 | fe 55°536 | 6-62 prrity * 
ii Aq | 529513 5.87 || 973580 | ay A; ap pled od 6.62 geo 14 
pe) of! 520018 5 8h - 973560 pe 55593838 oe .444067 | 13 
48 | 529864 BO || g7g585 | sf 556399 | 6.60 3671 | 12 
49 | ‘530215 | 5-85 || “o734e9 | 217 grid | 6.60 AA3071 12 
50 53065 | 5-88 “993 144 5 aD een tt AaOO 443275 | 11 
| . e 5.83 «dle | v4 I .557121 | 6 60 442879 Hl 10 
51 | 9.530915 | . o 9.973398 | 51 é 
52-| 53193 | 5-88 fens Pore pasts oh 6.60 peri : 
| 53.| 531614 | 3-82 "973307 | 23 |] “xsgaqgg | 6.58 441602 | 7 
54.| (531963 | 5-82 973261 | ff | ‘asgeqg| 6.58 441297 | 6 
55 | 532312 | 5-82 973215 | ee “s59097 | 6-54 4400081 LB 
56 | 539661 | 5-82 -g73ic9 | fe ‘50491 | 8-54 sip 4 
57 533009 5.80 || 973 wie! "5 1a ML |B By , 009 | 4 
Bg | 283008 | igo || files | Te 559885 | BP 440115 | 3 
: .05d500 | ny | .9f8018 | Arche 56027 “ts 43975 1 «ee 
59 | 1583704 | 5-48 || ‘orgose | 7% || “Beogeg | 6.5% ‘faeser | 4 
Mii 60 | 9.534052 | °-°? || 9'972986 | 77 || g'61066 | 6-55. | 10‘438934 | 0 
igocc coe «ee ree erro > | — 
t | Cosine. |.D.4".) ||) Sinew» } Dia*er| hea Tang. } 


| Cotang. 


ps | 
O° | 
ive) 
° 
~ 
© 
° 


378 


Sine. | D. 1". 


Cosine. 


COSINES, TANGENTS, AND COTANGENTS. 


Tang. 


ISO Wwe © 


9.534052 | 


ely, 
534399 | 5.78 
534745 | PR 
535092 | 248 
535438 | 5 pe 
535783 s 


.586129 
536474 | 
.536818 
5387163 
537507 


9.537851 


C9 OLED OT 2 OF 


III I-A 


3 
ina) 

.538194 | ~3 
538538 - 
538880 | 2 "ho 
(539223 | 2746 
589565 | 2-4 


539907 | 
£40249 
540590 
| .540931 
| 9.541272 | 
| 541613 


541953 oe 
542293 | O¢ 
542632 | 65 
542971 | 65 


.543310 
.543649 
.543987 
.544325 
| 9.544663 
.545000 | 
545838 
545674 | 
.546011 | 
546347 
.546683 
.547019 | 
5473854 
| 1547689 
9.548024 


+ 
© 


548359 | 2-28 
548693 | ee 
549027 | 22s 
549360 | Pu 
549693 | 2-22 
550026 | 2:22 
550359 ay 

vd 


550692 
.551024 


| 9.551356 


Orgor 


TON NNT OOO TOT OT OUT TTT OTT TOTO T TTT OTT LOTT TUN TUT TUTTE, SUITE OT OT IIIT UIT AN 
er : = ; She Mr Po a gu Cure thee 
oo (ou) 


eet 52 
551687 5 
a8 | See 
ee ope ¢ ve 
552849 5) 
552680 pe 
553010 =9 
553841 48 
553670 : 
~vv00 t 50 
.5_B4000 ae 

| 915 48 


54329 


9.972986 
972940 
972894 
972848 
972802 
x OT 275 = 


10d 


972709 
. 972663 
.972617 
.972570 
972524 
972478 
972431 | 
.97 2385 
972338 | 
972291 
972245 
.972198 
.972151 
972105 | 
. 972058 
972011 | 
971964 | 
971917 
.971870 
971823 
971776 
971729 | 
971682 
.971635 
.971588 
) 971540 
.971493 
.971446 
.971898 
.971851 
.9718038 
.971256 
.971208 
.971161 
.971113 
971066 
.971018 
970970 | 
970922 | 
970874 | 
970827 
077 
.970731 
970683 
. 970635 
.970586 
970588 
970490 | 
970442 | 
970394 
.970845 
970297 
970249 | 
.970200 | 
9.970152 | 


INQ D-III W-I-I-2-3 


By So a eo st a 


OW W-3 WOW O-32D 


@ 


Sag at) IES: 


| 9.561066 


.561459 
.561851 
562244 
.562636 
563028 


563419 


.563811 
564202 
564598 
564983 


| 9.565873 


565763 
.966158 
566542 
566982 


.567320 


.567709 
.568098 
.568486 
568873 


9.569261 
| .569648 


.5700385 
.570422 
570809 
571195 
.571581 
571967 
.5 72352 
572738 


| 9.573128 


578507 
575892 
514276 
574660 
575044 
ST5AR7 
.575810 
576198 
576576 
9.576959 
OV 7841 
57128 


578104 
.578486 


579629 


.580009 
. 580889 


9.580769 
.581149 


.582665 


5838044 
583422 
.5838800 


9.584177 


anne? 90909 OD Ww o kegs . 7 


> SD 


Cotang. 


10.488934 
.438541 
.488149 
437756 
.437864 
.436972 
.436581 
.436189 
.435798 
.4385407 


435017 | 


10.484627 


.434237 


433847 
433458 
433068 
432680 
.432291 
.431902 
.431514 
.431127 
10.480739 
430352 
429965 


429578 | 
429191 


428805 


428419 
428033 


.427648 
427262 
10.426877 
426493 


426108 


425724 


425340 | 
424956 | 


424578 
.424190 
423807 


423424 | 
10.423041 


.422659 


422277 


21896 


“491514 


.421138 
420752 
.420371 


.419991 


419611 


10.419231 
418851 


.418472 
.418093 | 
ALTT14 | 


417385 
.416956 
.416578 


.416200 | 


10.415828 


Het 
> “CO 


a 


ardradieed 
Go He Or 


+] 


we 


et 
_ 


10 


oahio) 


CS mt 09 CO HB OT S2 3 


‘ | Cosine. 


oO 


bets 


Sine. 


Cotang. 


=) 


Tang. 


Sine. Dl" | Cosine. | D. 1". | Tang. | D. 1". | Cotang. | / 
| 
| ; | | i a PM ae eae ek ee 
RRA Qe m 5 pyre | 
phate | 5.48 || 9-P0152 | gg || 9.584177 | aq | 10.415823 | 60 
“ -00FIG ¢ mk oun Ot De a 0049382 ne 41506 5 
3} .655315 | 3-4" | ‘970006 Bo || 885809 | 6:28 “4i4eo1 | 27 
4 |) 55568 | E47 || 969957 | 8° || pabese | 6-28 | . “ayant, | 56 
5) -So5071 | E47 || 969909) -89 || 586062 | 8-27 413938 | 55 
6 | 556209 | 545 || 969860 | 65 || 5864a9 | 8-28 | “args: | 54 
i) BES 28 | Me) S| Be) ge | tee 
eases igs -JOIIO 5 008 LE sated .41281 By’ 
oe | eee bo wae |] Soeeria | 8) Us eerbeg | 6-28 $1243 | Bi 
557606 | 5°43 || 969665 | 82 |) ‘oyioai | 8-25 412059 | 50 
Wi 11 | 9.557982 | 5 43 || 9.960616 | “4, || 9.588816 Foe | to-4tiess | 49 
Whey fe) .BOSB53 | Sp || -960567 | 82 || paseo | 6-25 ‘411309 | 48 
14 | 558009 | f'4o || 960469 | 88 || “beguz0 | 8-23 .410560 | 46 
15 | -Bpe2 | 5.40 || 969420) -g5 |] -580814 | 6-23 | latorg6 | 45 
| o} peeps | 5.42 || 269870) “> |) -s0188 | 6-33 | laoggie | 44 
i) Ti | epegees | Bug || peemet | Aes || Go0bee | 688 409438 | 43 
jit 18) -p60207 | 540 || 960272) +85 || -Bo0095 | 8-22 | ‘409065 | 42 
1 49) 560531 | 5'49 || 969223 | -g3 | -b91808 | 6-22 |  ‘4osege | 41 
Hi “D60855 | 5138 || 960173 |g) |} -doues1 | 33 | .408819 | 40 
i] vay om 5 ; 
"ie 21 | 9.581176 | 555 | 9. 969124 | go || 9-592054 | og | 10407946 | 2 
Ae He 23 | “561824 | 5-38 |] “gggo95 | 83 || “pyered | 6.22 pe 
vi 33 | -po1s824 | 537 || 969025 | --g) |] -b9ev00 | 6-22 | ‘aoreo1 | 87 
i) a ero | me? i a "99 598542 6.20 -406458 | 35 
4 36) b6200 | 537 || 968877 | -g5 |] -bosoa | 6-20 | ~dogose | 34 
i 2 | 5612 | Fy 968827 | G3 |] 594285 || 6-18 405715 | 83 
aM A Tee epee | Biay loeeeeere | tee || Spates || Bute 405344 | 82 
6) 2BBTS | Sieg |] 960728 |B [lc ponoe7 | B18 404973 | 31 
a 564075 | 5’a5 || -966678 | gs || 595808 | 6-18 | ‘4os6oe | 80 
il BI | 9.564306 | 5 a9 | 9. 968628 | gg || 9.595768] ¢ 47 | 10.404282 | 29 
f G2'| . ROATIG |. Brag || 968578 | 83 | pogias | 8.1? 408862 | 28 
Me 34. | (pebase | BeBB || 7eeebe8 | olga |] -896508 | g-te .403492 | 27 
H 35 | 565676 | 5-33 || “eggion | 188 || 7BP6STB | g'a5 | 408122 | 26 
Z . x ‘£4 5.35 ie Se) ae | ‘ SOU 12 Ry .402753 | 25 
6 ) 565905 | 255 || 90870 | 83 || ‘sore16 | 6-15 | “aopsea | 21 
eh lowipeeens 8oad Hae oe | 195 || -597985 | g48 .402015 | 28 
567269 | 5139 || 968178 | 7g 599091 | 743 -400909 | 20 
A swastowa OQ | : rm ; 
42 | eofee, | 5.28 || 9-968128 | “gy || 9.599450 | 1 | 10.400541 | 19 
42 | -p61904 | 5°59 || 268078 | G2 |] -sa9827 | 6-18 | «~“aoor7s | 1a 
ill] BB) pebee2 | peg ||: 908027 | (88 | “enon, | Bel2 899806 | 17 
Hi to | ceesee | Bier || 967927 | +g |) | 600020 | 6-12 |  “agg0rt | 15 
| 46 | -B6O172 | Siar |) 260876 | 58 || o1a0s | 6-12 | © “sonz04 | 14 
| 44 | 369488 | Sor || 967826] 23 || ‘601663 | 6-12 "898887 | 18 
| By | Uses | Bee 1 gee || cree 1] Sb0R0e8 | a8 397971 | 12 
iil 49 | 970120 | oe 967725 "Br 602395 | @” 897605 | 11 
| | 50 | -BM0S85 | 5:7 |] 907674 | 83 || Leon761 | 6-10 | “397099 | 30 
Bo Brows 3m Be Ps ; ; 
so | ceriogg | 5-25 || 9-2982624) gs |] 9.60818 | ¢ 49 | 10.396873 | 9 
53 | 1571380 | 5-28 |] “ogeton | 85 |] 808493) | g'og |  -896507 | 8 
a - os oes 595 : IOTSR2 g5 | 608858 | 6.08 . 896142 ff 
55 | 1572009 | 3-23 || 7pol8"1 | vg || -604228-1 ggg |  .B05z77 | 6 
fe cel” | Bias 967421 "a5 || 604588 | @'ng .895412 | 5 
Bo | 722828) 5ia0 || -967370| -g? || ecaona | 8-98 |  “Rosod7 | 4 
Bi 72686 | sag || 907819) 2 || “ecs3i7 | 8-07 |  “aqaess | 3 
Be] 7pie850 | pian || 967268] ge |] -eoscsz | 6-08 | “squgi8 | 2 
op | gpueee3 | 5.20 || 967217 "gs -606046 | 6 o> .893954 | 1 
0 | 9.57857 9.967166 “|| 9.606410 ™" | 10.893590 | 0 
Pine al Gaaeeke i are ee we; 
Rees Cosine. | D. 1 Sine. |! D. 1". |! Cotang.! D.1". | Tang. | / 


111° 


— 


D. 1", || Cosine. | D. 1". || Tang. 
= = es eG ae 
0 | 9.573575 | 5 99 || 9.967166 | es | 9-606410 
1| (573888 | 359 || -967115 | “ge || .606773 
2| .574200 | F'59 ||  -967064 "gx || -607187 
3 | .574512 | Foo || -967013 | “or || 607500 
4) .574824 | 559 || -966961 "Br 607863 
5 | :5751386 | - jg || .966910 "35 608225 
6 | 57447 | 54g ||  -966859 “85 608588 
7 | .575758 | £49 || 966808 | gn . 608950 
B | .576069 | P'y> || -966756 | “on .609312 
9} .576379 | Fyn || 966705 a 609674 
10 | 576689 | Fyy || 966653 "Bs .610036 
11 | 9.576999 | ~ 47 || 9.966602 ~ || 9.610897 
2 | .577309 5 15 .966550 "8s 610059 
13 | 577618 | 245 . 966499 “Be .611120 
14) BY7927 | P45 966447 eee .611480 
15 | 4578236 | 245 966395 "gs || -611841 
16 | .578545 | "43 . 966344 “2 ||. 812201 | 
17 | 578853 | 2-72 966292 | “So || .612561 
18 | 1579162 alee . 966240 pee . 612921 
19 | 579470 8 12 .966188 ee 618281 
20 | 579777 5 13 . 966136 "gs 613641 
21 | 9.580085 | ~ 49 || 9.966085 gy || 9-614000 
22-| 580892 | 5745 || -266033 rats 614859 
23 | .580699 | -'39 || -965981 “OF 1614718 | 
24 | 1581005 | 245 || 265929 “ag || -615077 | 
25 | 1581312] 2739 || -965876 |“, 615435 
26 | 1581618 | S39 ||  -965824 we 615793 
27 | .581924 | Fog || .965772 o 616151 
28 | 1582229] £49 || 965720 | || 616509 
29 | 582535 | F'og || 965668 “eg || 616867 
30 | .582840 | F*og ||  -965615 "gr || 617224 
31 | 9.583145 | ~ gy || 9.965563 | » || 9.61%582 
32 | 583449 | Pog || .965511 | “68 617939 | 
83 | .588754 | B'o~ || -965458 | > || .618295 
34 | 584058 | pion || -965406 | “GQ || .918652 
35 | 584361 | pp» || -965353 | “oe |) .619008 | 
36 | 584665 | 5" 05 965301 “oR 619364 
387 | .584968 | =o 965248 | “G8 619720 
88 | 585272 | z'o3 || -965195 | “gn 620076 | 
89 | 585574 | 5o5 || 965143 | ogg || -Sa0dse 
40 | a 585874 ( 5 ‘ 03 ie 965090 , 88 s 620 ( 8 ( 
4i | 9.586179 | x ox || 9.965037 | gg || 9.621142 
42 | .586482 | p'o9 || -964984 “ag || «| 621497 | 
43 | .586783 5 08 || 964931 ees! 621852 | 
44} 587085 | poo ||  -964879 | “6 622207 
45 | .587386 | p'o9 || -964826 “og 622561 
46 | .587688 | Foo 964773 “og 622915 | 
47 | 587989 | 2" aq .964720 | “gq || - 628269 | 
48 | .588289 | Py og || -964666 "oq || -623623 
49 -588590 5 00 -964613 | “a8 | 623976 
50 | 588890 | 2"o9 ||  -964560 “gg || -624330 
51 | 9.589190 , || 9.964507 | 19 ||: 9.624688 
52 | .5eode9 | 4-98 || “‘o64454 | “88 || -625036 
53 | .589789 | frog || -964400 | “ge .625388 
54} .590088 4.98 | -964847 | “ge 625741 
55 | .590887 | 4igg || -964294 “99 || + - 626093 
56 | .590686 | 4'g, || -964240 | “gg ||. 626445 
57 | .590984 | 4igy || -964187 | “99 || 626797 
58 | .591282 | 4ign || -964183 | “gg || .627149 
59 | 591580 | 4.97 . 964080 90 627501 
60 | 9.591878 Ai 9.964026 “|| 9.627852 
Y + Cosine: |°Dr1"7 ||; --Sine. Dra F11 Cotang. 


COSINES, TANGENTS, AND COVANGENTS. 


Dae 


6.05 
6.07 
6.05 
6.05 
6.03 
6.05 
6.03 
6.03 
6.03 
6.03 
6.02 
6.03 
6.02 
6.00 
6.02 
6.00 
6.00 
6.00 
6.0 
6.00 
5.98 
5.98 
5.98 
5.98 
5.97 
5.97 
5.9 
5.97 
5.97 
5.95 
5.97 
5.95 
5.93 
5.93 
5.98 


CS CS <O 
Co wo Oo 


OroTvOwor ar Ororororororororordr Ororororor 


i) 
~) 


Or or or 
3 


Cotang. | / 
10.393590 | 60 
,393227 | 59 
.892863 | 58 
.392500 | 57 
.892137 | 56 
B917%5 | 55 
891412 | 54 
.891050 | 53 
.890688 | 52 
.890326 | 51 
.389964 | 50 
10.389603 | 49 
.889241 | 48 
888880 | 47 
.888520 | 46 
.888159 | 45 
.B87799 | 44 
887439 | 43 
.387079 | 42 
.886719 | 41 

| .886359 | 40 
10.386000 | 39 
885641 | 38 
.885282 | 37 
884923 | 36 
.884565 | 35 
.884207 | 34 
.883849 | 33 
883491 | 32 
.883133 | 31 
382776 | 30 
0.382418 | 29 
382061 | 28 
881705 | 27 
.881348 | 26 
-380992 | 25 
.880636 | 24 
380280 | 23 
379924 | 22 
.879568 | 21 
.379213 | 20 
10.378858 | 19 
.878503 | 18 
.378148 | 17 
377793 | 16 
877489 | 15 
BU7085 | 14 
LOT Fol ako 
876377 | 12 
876024 | 11 
.375670 | 10 
10.375317 | 9 
874964 | §& 
874612 | 
374259 | 6 
373907 | 5 
BIB555 | 4 
.873203 | 3 
872851 | 2 
372499 | 1 
10.372148 | 0 
Tang. 


TABLE XXV.—LOGARITHMIC SINES, 


| ! 
Sine. D. 1". | Cosine. | D. 1”. || Tang. | D.1". | Cotang. | ’ 
| | | 
| 
0 91878 ~ || 9.964026 || 9.627852 ~ | 10.872148 | 60 
1| 592176 | 4-95 || 963072 | -29 || gase03 | 5:85 | “arr707 | 59 
2| .592473 | 49s . 983919 "90 |; 628554 | Pgs .871446 | 58 
3 | 592770 | 493 963865 “90 628905 | 239 371095 | 57 
4 | 593067 | 793 963811 "99 ||. 629255 | Boge 310745 | 56 
5 | .593363 | 793 963757 "gg || | 629806 | F "98 .870394 | 55 
6 | .593659 | 493 963704 "90 || 620956 | B*p5 .3870044 | 54 
7 | 593035 | 493 963650 "99 || 630806 | 5°35 .869694 | 53 
8 | .594251 | 493 || .963596 "99 || -680856 | 535 .869344 | 52 
9 | .591547 | 4io5 || 963542 "99 | 631005 | 2°38 .368995 | 51 
10] .594842 | 4g || .963488 "99 |, -631855 | S65 .868645 | 50 
i1 | 9.595137 | 4 99 || 9.963434 gg | 9-631704 | ~ g5 | 10.868296 | 49 
2| 595432) 495 || .963379 "99 || -682053 | ees .867947 | 48 
B} 595727 | 4°95 963325 ‘9p || 682402 5 80 867598 | 47 
14 | 1596021 | 4°95 963271 "99 || 682750 | 5 "65 .367250 | 46 
15 | .596315 | 4°90 963217 "99 || -688099 | 3 °p5 .366901 | 45 
16 | .596609 4g _«|| 963163 "92 || 688447 | 5°65 .366553 | 44 
7 | .596903 | 4°65 .963108 “90 633795 | 5 "6p .366205 | 43 
al 18 | 597196 | 4"o9 || -963054 ‘92 || -684143 | Bp .365857 | 42 
i) 19 | 5974901 4°93 . 962999 “90 -634490 | "Gq .865510 | 41 | 
Li 20 | 597783 gg || 962945 "92 || 684838 | She .865162 | 40 
Be 21 | 9.598075 4 go || 9.962890 go || 9-685185 | ~ no | 10.364815 | 39 
it 22 | .598368 ; 4'g- || .962836 "gn || 635582 | Bs .364468 | 38 
Nth 23 | .593660 “oy || .962781 "99 ||. -685879 | Ac .364121 | 37 
Hi 24] 598052 | 4:8" || 962727 oo || .636296 | 5-78 363774 | 36 
nt 25 | 599244 | 4°64 |) 962672 "99 || 686572 | Reng . 363428 | 35 
thi 26 | .599536 | Gig. || 962617 a 4680019 4) ore .363081 | 34 
27 | .599827 | 4 ‘oe 962562 “90 637265 | "mw 362735 | 83 
Hit 28] .600118 | 4g. || 962503 too || 687811 |) Ripe . 862389 | 32 
hal 29 | 600409 | 4g. || .962453 "99 || -687956 | oan .362044 | 31 
yt 30 | .600709 | 4'g3 || 962398 ‘92 || -638302 | Pes . 361698 | 80 
1H | 81 | 9.609999 | 4 gg || 9.962343 gg || 9.638647 | ~~ | 10.361353 | 29 
Baal | 32 | 601280 | 4°99 . 962288 aS -638992 | 5 ’ne .361008 | 28 
Hit 83 | .601570-| 4gq ||  .962233 i 639337 | 2 'np .360663 | 27 
ii 84] .601860 | Jigq || 962178 “99 639682 | 2 "ne .360318 | 26 
nt 85 | .602150 | 499 || 96212: “93 «|| «=~ 640027 B73 859973 | 25 
ei 86 | .602439 | fgg || 962067 | +05 || 640871 5 5 .359629 | 24 
Ha 37 | 602728 | 4'gg || .962012] 95 || 640716 5 3 .859284 | 23 
Pune: 38 | 608017 | 4°35 -961957 | 95 || .641060 | Pens .858940 | 22 
cial 89 | .603305 | 4'g5 || -961902 | 93 || 641404 5 72 .858596 | 21 | 
40 | 603594 | 4°35 . 961846 "gg || 641747 | rng - 858253 | 20 | 
41 | 9.603882 | 4 gy || 9.961791 gg |, 9-642091 | 5 ag | 10.3857909 | 19 | 
42] .604170 | 4p, .961735 "gn || 642434 | png 857566 | 18 | 
i 43) 604457 | 785 . 961680 “93 C4277 | Bing .857223 | 17 
i 44) .604745 | 4p 961624 “99 643120 | 2°15 .356880 | 16 
45 | 605032 | 4'n .961569 “98 643462 | 2°45 356537 | 15 
a 46} 605319 | 7+ 961513 a5 .643806 | 2-6 .856194 | 14 
| 7 | .605606 | 7"+- 961458 “93 644148 5 40 355852 | 13 
Ha 43 | -605892 | ging || -961402] “98 644490 | Pn5 .3855510 | 12 
Wnt 49 | 608179 | gin || -961846 | -95 || 1644832 | ?-f 355168 | 11 | | 
Hil! 50 | .600465 | 4's -961290 | “99 || .645174 | 2-6 .3854826 | 10 | 
51 | 9.606751 > 9.961235 2 || 9.645516 10.354484 | 9 
52 607036 | 4-72 || ‘961179 oa || 1645857 | 5.88 "354143 | 8 
53 | 07322 | Fins -961123 | “93 || 646199 |e "ha .353801 | 7 
b4} corer | 7-2 961067 | 93 || 646540 | 2-68 _353460 | 6 
55 | .607892 | 7 "ne 961011 "93_|| 646881 | 5 G0 -B53119 | 5 
| 56 | © .608177 493 960955 “93 |, 647222 5 67 352778 | 4 
57 | .608461 | 7°25 . 960899 "93 || 647562. ] Fe eos | 3 
BS} 608745 | 4-73 || 960843 | 92 || 647903 | 5-88 097 | 2 
59 | 609029 | fons 960786 | “93 648243 | Pope ‘35r73" | 1 
60 | 9.609313 “1 9.960730 | | 9.648583 | °°?" | 10.351417} 0 
7 Gosime.-| D241 Sine. PPE RAS 4 Cotang. | D. 1” Tang. | 1) 


Bine..-,|. Di", | Cosine. | D..1 | Tang. | D.1". ; Cotang | , 
aS 
= a 
0 | 0931: | ve 58: 
0 9.600818. 4-3. || 9.960780 | 93 9.648583. |, @ | 10.351417 | 60 
@ | asBOe80 | aia" <PEOGTS “gy |; 649263 | pgn | 800737 | 58 
B estaitse | aoae iL is 60561 | ‘93 |) .649602 | 5 67 .850398 | 57 
5} ceiory | 420 || “oeousa | 28 | Rees 5.65 oe |e 
Bl ernie | 40% |) CQeaaos |) f8r |! Seen || 5-85 BageT9 | 85 
6 012 | Zr. |] -960892 3 || [650620 |. 2: 349380 | 54 
| .611294 ‘ 960335 | “22 59 |, 9-63 “B49041 | 53 
8| ‘611576 | 449 "960279 ot ‘eoteo7 | 583 | “Sasros | Be 
Sears | oa | See | cae. | ih Se | aes 
| : 2 . 93 Vo ( rR € . 348026 50 
11 | 9.612421 | | o.e5312 | 
12 | venroe | 4:53 || °Sooos9 | 93. |) epans0 | 8-63 | 7-Betaso Fe: 
33 | ce1a9es | 4-68 |] “osgo95 | 95 |! “oppose | 8-88 | cBatore | a7 
| cerae6s | 453 |) “gpg | 93 || cesaaen | 8-63 | “Stgora | 46 
15 | 18545 4-68 || ‘ososse | 783° || 658663 | 5.62 | “Bieeee | 45 
613825 , 4°00 59825 | “22 |! <7654000 |; 5-82 
17 | cori | 48% || cSsomos | “23 | epaaar 882 | “Bases | 43 
18 | “614385 | 4°8% || “osovi1 | °9 |) ‘esaerg | 5-62 pret 
9 | me lt aie tate be = || -654674 ee .345326 | 42 
o | 614065 | 65 |] 939054 sey || Eeoeenit st Bake "344980 | 41 
oi eto 4.65 ae 05 655348 | 560 .044652 | 40 
S| Grae | 4.65. || 2ENES | 95 || *bep5090 | 5:60 | 7° Ses9a0 | Bs 
93 | eisrs1 | 4-8 || ‘osq405) -25 || “@5635g | 5-60 Serie | ay 
S| ce1eoeo | 455 || gsyses | 225. |! “eseone | 5:80 | “Biss08 | 86 
| ceiosay | 4-63 || “gagso | 9% |] ceoroas | 5:60 | “Sisore | Bs 
96 | 616616 | 4:63 || ‘959953! 25 || “G57gea | 5-80 "Baeese | BA 
a | ceiesoe | 463 || cgpyros | 22 |) ce8tosa. B88 | “brpsor | 83 
98 | ¢i7i72| 4:83 || ‘osgisg| <2 | ‘gosoat | 5-58 Bito06 | 32 
29 | 617450 4.63 | “g59080 | -2% |] ° \assaco | 3-58 Su1031 | 31 
‘ C17797 -0# Kans oot Re, : , 
eure | 4:82 || “959023 | -29 || lessvos | 2°33 |  -841296 | 30 
a 9.618004 3 an 9.958065 95 || 9-659039) 5 x | 10.340961 | 29 
2| 61881 | 4° "958908 | 25 ||° 1659373 | 2 340627 
33 | 619558 | 4:52 || “95885 ‘97 |) “E5q708 | 5-58 Sites | 
Bt | “eres | 4-60 |) “deeroa | -9% |! “eeoos | 9-32 eaeee (28 
35 | 619110 4.60 || 958734 | <2f || 660376 | b,0% ees | 35 
5 | “e19386 | 4: ‘958677 | 22 || ‘eeovi0 | 2-52 |  ‘3sgg00 | 2 
37 | 619662 | 4:80 5 ‘OT | “Be 1o43 | 5-55 ener te 
| Giooss | 4-50 |) capssor | 2% |) copier | 5:57 | “Sapo | 22 
BF | cebeers | 458 || “pores | 2% |} ceerrio | -5:55 | ‘Baso00 | 31 
[ -oandss | 53 |) 2585 | oor 662043 | >>? 37957 | 20 
41 | 9.620763 | 5g |) 9.958387 | g, |) 9.662876 | “~~ | 10.337624 | 19 
2| 1621038 | 4° "958329 | 2% ||. ‘62709 | 5-55 | © ‘aava9 
43 | 1621313 | 4°oo || coseevl | <4 pean | 5:55 | “B3g099 | 47 
44} 621587 | 472 ‘osseia | 2% |) “eusa7s | 5-55 Sets tee 
Biker | 4:57 |] cQeeise | 98 |) cGoaror | 8-23 | “Bsoe08 | 15 
46 | .6e2ias | 4-27 || ‘958006 | 2 || “664039 | 5-33 ‘S35001 | 14 
7 | 1622409 | Gree 58035 OF 664371 | 2:38 "33562 
Br (ieee | Ace i tiers || 208 || criers | 5 :b8 | Bate, Ee 
PP ieee | iis, leet | ibe |i cceeayOS | pbs | cater me 
ey cere | 4a5b [oe ieee | ct || -665035.) 559 834905 | 17 
4.55 ~dd (ODD 98 . 665866 | kK 53 .304634 10 
51 | 9.628502 9.9578 || 9.665 oe. 
Bl | 9.628502 | 4g nq ||,9.057804 |g, || 9.085608 | 5 59 10.334302 | 9 
52 | .oaur7d | fps |) 980746 | <3 || 600% | bio | -BaBOTL | 8 
BS | 624047 | 43 || -957687 | “gg || - 666360 | es .333640 | 7 
Bd |r s624319 | 453 || ee “OF 666691 | 2°25 .333309 | 6 
56 | 624863 | 258 || ‘osvai| +28 “gorse | 8-52 8 | a4 
57 | 605185 | 4°28 || opvene : «28 “porees | 3-00 | “bammg |B 
58 | ‘62540 | 4-22 |] “osvao3 | 28 || “eeg013 | 5-52 aiiay | Se 
59 | 625677 | 4°22 || ‘os7Ba5 | -24 || “osasaa | 3-30 spiosy | 
2 age vn yrrore . : nisi 2 y | 
60 | 9.625048 |" 9.957276 9.668673 | 5-59 | 49'331397 | 0 
Gosine. |. D; 17; || ~Sines } Dil Cotang. | D.1". Tang. ear 


COSINES, TANGENTS, AND COTANGENTS. 


10 


Sine. 


9.625948 
626219 
. 626490 
. 626760 
627030 
627300 
627570 
.627840 
628109 
.628378 
628647 

9.628916 
.629185 
629453 
629721 
.629989 
.630257 
630524 
.630792 
-631059 
6313826 


9.631593 
.631859 
632125 
. 632392 
. 632658 
632923 
. 633189 
633454 
633719 
. 633984 


9.634249 
-634514 
63477 
.635042 
. 635306 
. 635570 
.635834 
. 636097 
. 636360 
636623 


9.636886 
.637148 
637411 
.637673 
637935 
638197 
638458 
638720 
638981 
639242 


9.639503 
.639764 
. 640024 
. 640284 
.640544 
. 640804 
. 641064 
.641324 
.641583 
9.641842 


Cosine. 


i 


1 


HB Orororor 


wwwwoesweasS GRGRARASLS SBRRSSSSSSIR 


SP RP 


¢ Ww CIt co CDWWKWHRA HR AHH 
GLOUNESITNINENIDON DODDOOOSOCOW 


co GY Co 


ALA AA AL AAA AAA ADA AAA AAA AD AAA RAAA AA AAA AAA RADA AAAAAARARAA 
Gt 6559 Or Or 


aad 
Co Cs 
02% 


Cosine. 


9.957276 
957217 
957158 
957099 
957040 
. 956981 
956921 
. 956862 
.956803 
956744 
. 956684 

9.956625 
956566 
956506 
956447 
956387 
956327 
956268 
. 956208 
. 956148 
. 956089 


9.956029 
. 955969 
. 955909 
. 955849 
955789 
. 955729 
955669 
. 955609 
. 955548 
. 955488 


9.955428 
955368 
. 955307 
955247 
. 955186 
955126 
955065 
. 955005 
954944 
954883 


9.954823 
. 954762 
954701 
. 954640 
. 954579 
954518 
954457 
- 954396 
954335 
954274 

9.954213 
- 954152 
-954090 
954029 
. 953968 
. 953906 
953845 
. 953783 
. 953722 

9.953660 


Sine. 


Dias 


bet eR eh ee pe ee ee pe pe pp | ell eel geet eel eel eee ee ed beh peek | tl eel and 
he one ape are ore he. He ime ate th see Ais may a ee pdt oale ai, Sori She ote ak yak oe yas Tye Se Sal ee GA ce e 2 ys 20 Ces Sea OA 

> 

i=) 


p| 
| 


| 9.668673 


TABLE XXV.—LOGARITHMIC SINES, 


Tang. 


. 669002 
. 669332 
-669661 
.669991 
670320 
670649 
670977 
671806 
.6716385 
671963 


9.672291 
672619 
672947 
673274 
673602 
673929 
674257 
674584 
674911 
675237 

9.675564 
675890 
676217 
676543 
676869 
677194 
677520 
677846 
678171 
678496 

9.678821 
679146 
679471 
679795 
680120 
680444 
680768 
681092 
681416 
681740 


9.682063 
682387 
682710 
. 683033 
. 683356 
. 683679 
684001 
684324 
.684646 
. 684968 


9.685290 
.685612 
- 685934 
. 686255 
686577 
. 686898 
687219 
.687540 
.687861 

9.688182 


Cotang. | 


D. 1’. 


5.48 - 


LPL PPR PP PP Sh PP PP 
SSSSSSERS5E5 fo fo to Ge Oo wo Go to Or Go Ot OW OF I ON SS3ea55 


HAAN NNO OOOO HOON TON TOT OUT, CLOT OT ON OT OT OT OT OTT. OTOTOTOTOTON 
PLLA DLP PPA . 8 Pa SE te Te War Ban 


TOLOTOVOUN OTSI AE AE 


OCeOLOvr gor orgorvororor O1ror 
a7 re aera ee Co OO 


o 


Cotang. 


10.331327 
. 880998 
. 830668 
330339 


. 830009 
. 829680 
829351 


829023 
828694 
828365 
828037 


10.827709 
.327381 
.3827053 
326726 
826898 
.826071 
820743 
820416 
. 325089 
824763 


10.324436 
-824110 
828783 
823457 
.823131 


822806 


822480 
- 822154 
-821829 
821504 


10.321179 
820854 
820529 
820205 
319880 
319556 
819282 
. 318908 
. 318584 
.318260 


10.317937 
.3817613 
317290 
3816967 
.316644 
£816321 
. 3815999 
-815676 
-815854 
-815082 


10.314710 
.314388 
.314066 
.818745 
813423 
. 813102 
312781 
312460 
.8121389 

10.311818 


Tang. 


| 


et 


~ 


COnt Co Ot te C52 


COSINES, TANGENTS, AND COTANGENTS, 


Sine. Diss 


9.641842 
"642101 reo 
642360 | 4-32 
642618 q's3 
642877 | 4-32 
"643135 e+ 
643393 | 4°30 
613650 | 4°55 
.643908 4 28 


644165 , 
RA AADE 4.30 
644423 4°28 


| 9.644680} 4 o» 


"644936 | 4 
"645193 ye 
"615450 


6550: 


. 655805 

. 656054 

656302 

.656551 

.656799 | 
9.657047 


or 
645706 a 
645962 | 7°56 
646218 427 
646474 1.9% 
646729 | 4°95 
646984 | 4°5> 
Garad | 423 
647749 het 
648004 | 4°53 
648258 | 4°53 
648512 | 45. 
648760 | 4°5: 
649020 | 4°53 
649274 | 7°55 
649527 | 4°55 
9.649781 A 
“650031 | 4°32 
650287 | 7°59 
650539 | 4°55 
650792 | 4°55 
651014 | 4°55 
651297 | 4°59 
651549 | 4°75 
651800 | 4"op 
“652052 | 4° 
9.652304 mi 
.652¢ 
658555 a 
652806 | 4°48 
653057 | 449 
653308 | 4‘d> 
653558 | 417 
653808 418 
"654059 | @:18 
654309 reel 
654558 | "4p 
4 
4 
4 
4 
4 
4 
4 
4 


Cosine. | D. 1’. 


Cosine. 


9.953660 
. 958599 
-958537 
. 953475 
.953413 
- 953352 
- 958290 
. 953228 
- 953166 
.958104 
. 953042 


| 9.952980 


952918 
952855 
952793 
952731 
952669 
952606 
952544 
952481 
952419 
952356 
952294 
952231 
952168 
952106 
952043 
951980 
951917 
951854 
951791 


9.951728 
951665 
. 951602 
.95158 
951476 
.951412 
951849 
- 951286 
951222 


.951159 


co 


. 951032 
950968 
950905 
950841 
95077 
950714 
950650 
. 950586 
950522 


9950458 


. 950138 
. 950074 
. 950010 
.949945 
9.949881 


Sine. 


9.951096 


950894 

950330 | 
950266. | 
950202 | 


D1"; 


Tang. 


9.688182 
. 688502 
. 688823 
.689143 
. 689463 
.689783 
.690103 
.690423 
.690742 
.691062 
.6913881 


9.691700 
. 692019 
. 692358 
. 692656 
.692975 
698293 
.693612 
693930 
694248 
.694566 


| 9.694883 
.695201 
.695518 
.695836 
.696153 
.696470 
696787 
.697103 
.697420 
.697736 

9.698053 
| .698369 
. 698685 
.699001 
.699316 
. 699632 
.699947 
. 700263 
700578 
. 700893 
701208 
.701523 
.701837 
702152 
702466 
. 102781 
. 708095 
- 703409 
103722 
704036 
| 9.704350 
| .704663 
. 704976 
| .7'05290 
. 405603 
~ 705916 
706228 
706541 
| .'706854 
| 9.707166 


© 


| Cotang. 


Dsl"; 


Orororor 
) OO 
© 


CrOLCOvrOrorwore 
eo a 
i) 


OCvrOT Or or or 
ry) 
So 


5.382 


CUNO UOT OT OT Oot OT OT OU OU St OF OF OF OF OF 
ra) 
(SC) 


| 
| 


Cotang. 


10.311818 


| 


-811498 | 


811177 
810857 


810537 | 
310217 | 
309897 


809577 
-809258 


808938 | 


. 808619 


10.308300 


807981 
-807662 
807344 
807025 


306707 


- 806388 
806070 
805752 
805434 


10.305117 
. 804799 
. 804482 
. 804164 
. 803847 
.008530 
et BUGR IS 
- 802597 
. 802580 
.002264 


10.301947 
-801631 
.801315 

* 300999 


.800684 | 
300368 | 
800053 | 
299737 


299422 
299107 


10.298792 
298477 
298163 
297848 
297534 
291219 
296905 


296591 


296278 | 


295964 


10. 295650 
337 


ee 


a 


istatoa 
.294339 ¢ 


294084 | 


2932 
-298459 


.293146 


10.292834 


Tang. 


, 


Sine. Cosine. 
0 | 9.657047 | 4 43 9.949881 
1} .657295 | 449 949816 
2| .657542 | 445 949752 
8 | .657790 | a9 949688 
4 | .6580387 | 4745 949623 
5 | .658284 | 445 949558 
6 | 658531 | 4745 949494 
7 | 658778 | 445 949429 
8} .659025 | 4°49 949364 
9 | .65927 410 . 949300 
10 | 659517 | 4°49 949235 
11 | 9.659763 | 4 49 || 9.949170 
12 | 660009 | 4°49 949105 
13 | .660255 | 4°49 949040 
14] .660501 | Jog 948975 
15 | 660746 | {og 948910 
16 | 660991 | 4p. 948845 
17 | 661236 | 49g 948780 
18 | .661481 | 4 "og 948715 
19 | 661726 | Gop . 948650 
Q 661970 407 . 948584 
21 | 9.062214 | 4 og 9.948519 
22 | 662459 | 4" o 948454 
23. | 662703 4.05 948388 
24 | 662946 | 4 oe 948323 
25 | .663190 | 4"ox 948257 
26 | .663433 1.07 948192 
27 | .663677 1.05 948126 
28 | .663920 4. 05 948060 
29 | 664163 | 7p: 947995 
30 | .664406 4.08 947929 
31 | 9.664648 4.05 9.947863 
32 | .664891 4.08 947797 
33 | 665133 | 7 "o3 947731 
B4 | .665375 | 49g 947665 
35 | 665617 | 4°93 947600 
36 | 665859 | 4'o5 947533 
37 | .666100 | 7 "oa 947467 
38 | .666342 402 947401 
39 | 666583 | 4 "95 947335 
40 | 666824 | 4'o9 947269 
41 | 9.667065 | 4 9 9.947203 
2 | 667205 | 495 947136 
43 | .667546 | 4"p5 947070 
44 | 667786 | 7° 947004 
45 | 668027 | 4'o9 946937 
46 |  .668267 | 3*9g || -946871 
7 | .668506 | 4'o9 || 946804 
48 | .668746 | 4'o9 || -946738 | 
49 | 668986 | S'o3 || .946671 
50 | .669225 | 3°98 || .946604 
51 | 9.669464 | ,, 98 || 9.946538 
52} .669703 | 5°99 || .946471 | 
53 | 669942 | 5*98 946404 | 
54 | 670181 | 3 '" 946337 | 
55 | .670419 | 39g 946270 | 
56 | .670658 3°97 . 946203 | 
57 | 670896 | 3 *o. . 946136 
58 | .671134 3°97 946069 | 
59 | 671372 | 3 'o¥ 946002 | 
60 | 9.671609 “e 9.945935. | 
’ | Cosine. Dslr Sine. 


Dx? Tange 18D,A". Cotang. 
Hog || BARES br ogtog | ieee 
1.07 || crorzg0 | 5-2 292210 | 
tog. || -70st02 | 8°20. | © “Sareog | 
1.08 || -/0Stlt | 5.20 eee 
1.07 || “pooa7 | 518° | “Soa63 | 
1.08 "709349 | 3-20 290651 
1.08 || “vo9e60 | 5-18 "290340 
pe || Lemmag7t |. Bate "290029 
poe -7ion62 | 2-18 989718 | 
hank 5) » 
08 || Sipongs | Onde | “aeseeang | 
1.08 “711215 | 5-18 “O88785 | 
1.08 711595 | 2-17 “ORRATS 
1.08 "711836 | 2-18 "288164 
1.08 “712146 | 2-17 "987854 | 
1.08 “712456 | 5-17 O87 44 | 
1 ° 08 ; ay 2786 5 . WV 3 987234 
10° || cvago7e | 2-27 |  Topegeg 
108 713386 | 5°49 286614 | 
1.10 eyagia | 5-15 "O85 686 | 
1.08 14604 | 5-17 "985876 | 
1.10 "714933 | 5-15 "O85 067 | 
1.08 || cised2 | 3-15 | | lopaiss | 
1.10 715551 | 5-15 "984449 | 
ioe W15860 pe "984140 
08 716168 | 5-1: "983839 | 
10 "716477 ee 1983593 | 
716785 9715 
rio! || Catiebee | Bele | eens | 
340. || Satan | ae "989599 
1.10 “7iv709 | 2-2 "989291 | 
1.08 m1goi7 | 5-13 281983 | 
t8for || asztees | Baas 981675 
1-10 118038 | pos .281367 
J "719940 | 9-12 "981060 
ager || Setaeas | B38 "980752 
110 719555 | 2°45 .280445 
9.71986 28018 
1.12 || “stooieg | 5-12. | 77Seoest 
1.10. || “0476 | 5-18 "970524 
1 . 10 : Pores | 5 ° 12 ‘ 979917 
1.12 || “rer0a9 | 5-10 | “oregon 
1.10" || “igeia0e | 5-12, | tepgeed 
1.02" || o7et702 | 5-18. | “Te7eeos 
1.10 aeeeans |i Boke dbeies 
tag || ezeeatd | 8-49. | lopreas | 
110 T2221 | yg | 207879 | 
toe! | Teese | bape) Aas 
112 |] pass | Pay | 376462 | 
= 2 1] gee o. Weare Es. 
132) |) ae | 508 see 
The “Od A | 5.08 | "OWBB46 
1.12 "94760 | 5-10.) ‘oenoan 
1.12 ‘wonogs | 5-08 “O74935 
1.12 Liste “0 5.08 974830 
1.12 caeuel 5.07 peer 
‘12 || go 7osera | 5: 1027432 
D: 1", {I Cotang. | D. 1’. | Tang. 


vo) 


OM dD wR OIE 30 


~ 


for) 
9 | 
° 


COSINES, TANGENTS, AND COTANGENTS. 


, Sine. D. 1”. 
Je le a 
0 | 9.671609 | 4 o, 
1| 67187 | 3.97 
2| 672084 | 3-9 
3 | 672821 | 3.95 
4| 672558 | 3-95 
5.| .672795 | 3-95 
6 | .673032 | 3-95 
7 | .673268.| 3-98 
8 673505 3 3 
| 673741 | 3-83 
10 | pane 393 
11 | 9.67421: 
12| .674448 ae 
13) 674684 | 3-88 
14 | 674919 | 3-92 
15 | 675155 | 3-98 
16 | .675390 | 3:92 
7 | 675624 | 3-90 
18 |, .675859 | 3:92 
19 | 676094 | 3.92 
20 | .676328 | 3-5 
* % 6763 : 3.90 

21 | 9.676562 
22 | 67796 | 3-90 . 
23. | (677030 oor 
24 | 677264 | 3:20 
25 | .677498 | 3:00 
26] .e777si | 3-88 
27 | .677064 | 3-88 
28 | .67si97 | 3-88 
29 | .678430 | 3-88 
30 | 678663 | 5°68 
31 | 9.678895 ie 
30 | B79128 3.88 
33 | .679360 | 3.8” 
34} 679502 | 3°» 
35 | .670se4 | 3-87 
86 | .680058 | 3°20 
37" .680288 | 3-87 
38 | .680519 3 Bs 
39 | .680750 | 3-8 
2AnARD | OY. 
40 pene 3.85 
41 | 9.681218 
42 | 681443 | 3.53 
43} .681674 385 
44] .681905 | 3°93 
45 | .682135 | 3/83 
46 | .682365 | 3-83 
47 | 682595 | 3-82 
48 | .682825 | 3-82 
49 | .683055 | 3°85 
= 209909 -¢ 
50 | 688284 | 3-88 
51 | 9.683514 | 2 a, 
52 | 683743 | 3-05 
53 | 683072 | 3-82 
54] 684201 | 3-88 
55 . 684430 3 ; 80) 
56 | .684658 | 3°35 
by | .684887 | 3°05 
58 | 685115 | 3°85) 
59 | 685343 | 3°5) 
60 | 9.685571 | 2 
/ Dit 


Cosine. 


Cosine. | D. 1’. Tang. D. 1".-| Cotang. 
9.945935 ; 9.725674 10. 274326 
945868 | 1-12 || l725979 oe 274021 
$1500 | ire || “BOM | sir |) 2reH16 
. 945733 1 : 12 ; (26583 5 07 213412 
945666 | 4°43 726892 | 2" 6s .273108 
945598 | 3°45 27197 | "op .272803 | 
945531 | 5°55 20501 | 3 *oe .272499 
945464 | 375! 727805 | Pye 272195 
945396 | 343 728109 | 5 "ox .271891 
945828 | 3775 (28412 | 2° op 271588 
945261 | 3°33 . 728716 5 Oy 271284 
9.945193 | , ,, || 9.729020] _ 10.270980 
945125 | 7-13 || ° "729393 Pee 270677 
|} 945058 | 3°33 ..729626 5 05 270874 
944990 | "45 . 729929 5 07 270071 
Qoo ote KO MONE . POVRYY 
944786 13 |) crgoses | 5.5 969162 
944718 | 3°45 sata || ete 268859 
-944650 | 4°33 731444 | Poo9 268556 
944582 | 3745 731746 | "og .268254 
9.944514. | | 9.782048 | _ 10. 267952 
stauie | 148 |) rs | 3.05 | reso 
toga | 1d8t ||) Oeoeee (2 Saas || Saat 
“O44: 143+ || Shaper | 5208 itis 
944241 115 (88257 5 02 266743 
WAIT? | S49 733558 | "93 -266442 
-944104 | 3+43 || .738860 | F°o3 .266140 
944086 | 45 || .784162 | P95 .265838 
948967 | 3°73 (34463 | o5 .265537 
943899 | "45 734764 | P'o3 £ 265236 
9.943830 ‘i: 5 || 9.735066 | o9 | 10.264934 
943761 | 3°45 735367 | 299 264633 
943693 | 55 735668 | 299 .264332 
943624 | 3773 735969 | 299 264031 
.948555 115 736269 | Foo 268731 
-$43486.| 4°95 |] «786510 | 5G |. - 263180 
angie |e AD || hgeke || Bs02e PE gener 
eeeee> Hy a, | eee | 5 60 ee 
948279 {| 15 . (37471 5 00 -2OROR9 
943210 | "75 BM | 269 262229 
| 9.943141 | | 9.738071 10.261929 
oin07e 4 14E || “zagazt | 2-00 261629 
. 943003 145 |) -88671 | Fog .261329 
.942934 147 | -¥38971 | 509 -261029 
Soon i iae ¢\|, Seeneen (oe | Pee 
942795 1,13 | (395700 5.00 me ‘ 
. 942726 1 “47 | . (39870 4.98 -260130 
942656 | "ys || -740169 | 4'98 259831 
942587 | "77 || 740468 | Fog .259532 
M2517 | 45 140767 | 49g . 259233 
9.942448 | 1 yn || 9.741066] 4 og | 10.258934 
942378 | yyy || -741865 4.98 258635 
-942308 | 3 "15 741664 | 4 "9p 258336 
|} 942239 | "4° 741962 | 498 . 258038 
942169 | 5 "y0 742261 | 4 ge 257739 
942099 | "yn || -742559 | 49g 257441 
-942029 | syn || -742858 | 4 gn 257142 
|| -941959 | 44 743156 | 4» ee 
Whee Serer ieee brea | Pe ee a 256546 
9.941819 9.743752 10. 256248 
| | | | | 
Sine. Dey Cotang. | D. 1". Tang. 


Dr COP O13 OO 


~ 


{ 


top] 
_ 
° 


Sine. | 18 A hase | 


9.685571 
685799 
686027 
686254 
- 686482 
686709 
. 686936 
687163 
687389 
.687616 
687843 


9. 688069 
688295 
. 688521 
688747 
688972 
.689198 
689423 
.689648 
.689873 
.690098 


9.690323 
690548 
690772 
690996 
691220 
691444 
691668 
691892 
“692115 
692339 


. 692562 
692785 
.693008 
.693231 
.693453 
693676 


ive) 


. 693898 »}: 


694120 
. 694342 
694564 


9.694786 
.695007 
695229 
.695450 
695671 
695892 
.696113 
696334 
.696554 
696775 

9.696995 
697215 
697435 
.697654 
697874 
. 698094 
.698313 
698532 
698751 

9.698970 


Cosine. 


3. 


Fates 


-~3 +3 3 
OLOTOre Or 


STAFF a st st ss 


Wo Wewowowwwwo 


68 


OCTOCOHN OTR NIEOT SIA RIO ND 


Cosine. 


9.941819 
941749 
941679 
.941609 
941539 
941469 
- 941398 
941328 
. 941258 
- 941187 
941117 


9.941046 
940975 
. 940905 
940834 
.940763 
940693 
940622 
940551 
940480 
. 940409 


9.940338 
- 940267 
. 940196 
. 940125 
. 940054 
. 939982 
939911 
. 939840 
939768 
. 939697 
939625 
939554 
939482 
. 939410 
939339 
939267 
939195 
. 939123 
- 939052 
. 938980 


9.938908 
938836 
928763 
938691 
. 938619 
938547 
938475 
933402 
938330 
.938258 


.938185 
. 938113 
. 938040 
937967 
937895 
. 937822 
937749 
937676 
. 937604 
9.937531 


Js) 


oO 


D. 1". 


oo 


o 


=| ARAAAAGIAGS Agora 


Sine. 


TABLE XXV.—LOGARITHMIC SINES, 


Tang. 


9.743752 
744050 
744348 
744645 
(44943 
745240 
745538 
145835 
(45182 
146429 
- (46726 

9.747028 

747319 

147616 

T4918 

748209 

(48505 

748801 

449097 

. 149393 

. 749689 

(49985 

750281 

150576 

. T5087 

(51167 

. 751462 

01757 

(92052 

192347 

. 152642 

9.752937 
158231 
(03026 
153820 


=) 


Jo) 
or 
» SO 
es) 
a 
S 


(59687 
159979 
160272 
760564 
. (60856 
761148 
9.761489 


Cotan ge. 


D. 1". | Cotang. 
10. 256248 
rp 955950 
ae "955652 
ye "955355 
re "955057 
= “254760 
re "954462 
ae 954165 
ae 953868 
ae "958571 
10 .252977 
pe "852681 
re "252087 
be "951791 
4th "951495 
a 251199 
cae 950908 
‘ik “250607 
ri "950811 
10. 250015 
bit "249719 
dike "949494 
ri "249198 
aes 248833 | 
re "948538 
Ae 248243 
re 247948 | 
oe "247653 
10.247063 
re 246769 
re "246180 
ye "945885 
br "945591 
re "945207 
pe 245008 
7a "241709 
it "O44415 
10.244199 
ie "343998 
re "843535 
ie 243241 
4.88 242948 
4 7 88 3 242055 
re "949369 
‘ee 242069 
re ‘Ont 
re "941483 
~ | 10.241190 
‘e 240898 
ie "240605 
4 "940313 
4a "240021 
4 4 87 . 939728 
ee 939436 
aa "939144 
rs "238852 
‘85 | 40'938561 
D: 1 Tang. 


/ 


SrKHwmOWhROIOIMHO 


~ 


oO 


BO A3.C> OT C9 2D 


Sine. D. 1”. || Cosine, | D, 1’, Tang. 1D ge Cotang. | / 
9.698970 ; || 9.937581 9.761439 0.2385 
60189 | 3-83 |) oazasg | 1-22 |] “verzai | 4.82 rears 
699407 | 3"Re 937885 | 7°38 “7En023 | 4:84 237977 | 58 
090626 | $"ga || .9a7312 | 1:22 || cyeeaia | 4-85 237686 | 57 
699844 | 343 || .937238 | 7-33 || 762600 | 4-87 "237304 | 5G 
700062 | 3°63 937165 | j'oo | || seape7 | 4785 237103 | 55 
700280 | 3°85 || <o3zo92 | 1-22 || ‘7esigg | 4-85 936812 | 54 
700498 | 3°43 || .937019 | j-e5 || zeaav9 | 4-85 236521 | 53 
-700716 | 3"¢5 || .936946 | j-Sq || .zea770 | 4-85 | “aaeps0 | 52 
700983 | 3°95 || .936872 | j-S || :ve4o61 | 4:85 235939 | 51 
701151 | 3'g9_ || 936799 | 393 ||) .7es352 | 4-85 | “235648 | 50 ! 
9.701368] 5 go || 9.936725 | 5 o9. || 9.764643 | 4 95 | 10.235357 | 49 
W1585 | 3°go || .936652 | j'55 ||. 763933 | 4-83 "235067 | 48 
701802 | 3'g5 || 936578 | 565 ||, 765224 | 4:85 | loaarve | a7 
702019 | 3"B5 936505 | 795 ||’ 765514 | 4-88 234486 | 46 
702286 | 5°gy || 936431 | 4-53 || i765805 | 4-85 "834195 | 45 
102452 | 3'go || .936857 | 1-53 || ‘766005 | 4:88 | lessons | 44 
702669 | 3°go || 936284 | j-55 || “76s285 | 488 |  losseis | 43 
702885 | 3'gq ||’ .936210 | }-53 || <vese7 | 4-88 "933305 | 49 
703101 | 3 Eq 936136 | "55 7166965 | 4°G8 "233035 | 41 
708317 | 3°69, || -936062 | 4°53: || .767255 | 4-83 232745 | 40 
9.703583 | 3 gq || 9.985988. | 1 93 || 9.767545:| 4 49 | 10.282455 | 39 
703749 | 3'zg° || .980014 | 1-58 || “veresa | 4-82 "232166 | 38 
703064 | 3'5g || .985840 | j'p3 || .768ie4 | 4-83 |  ‘o3i876 | 37 
704179 | 3°69 || 935766 | t-53 || -ves4id | 4-88 "931586 | 36 
-704395 | 3°53 || .985692 | j'53 ||° .768703 | 4-82 231297 | 35 
701610 | 32g || .935618 | jd, || 768002 | 4-82 | ‘931008 | 34 
701825 | 3"pe' || 995543 | 1-62 ||: czeoz8i | 4-82 "230719 | 33 
205040 | 3'57 || 985469 | jog || 769571 | 7-83 | ‘230490 | 32 
705254 | 35g || .985895 | j-5e || .7e0860 | 4-82 "930140 | 31 
705469 | 3°p7 || 935820 | 1-53 || .770148 | 4-80 '229852 | 30 
9.705683 9.935246. | 4 o~ || 9.770437 , | 10.229563 | 29 
“705808 | 3'Py || .oai7t | 1-28 ||“ \vrora6 | 4-82 | ‘gpoerd | 28 
706112 | 3'57 || .935097 | ‘ox P01 | 2 4'e5 | 828985 | a7 
706826 | "px || .935022 |. 3-53 rene | ee 228697 | 26 
“706589 | 3"po || 934948 | 755 Tribee.| | 788 "228408 | 25 
906753 | | 57 || 1934873 |. } Se 771880 | | 4:80 "228120 | 24 
706967 | "Px || 934798 | J "oe 772163 | 7°80 227832 | 23 
707180 | g:pe || 2984723 |. 1-53 TI2457 | | Fos 227543 | 29 
707893 | 3 pe || 1984649 | j-S5 || <7re74s |. 4-20 "227255 | Qt 
.707606 | 3°55 934574 | 5 "on 173033 | 4" 99 .226967 | 2 
9.707819 | 2 xx || 9.934499 ~ || 9.178821 10.226679 
oso | 3°72 || .934aza | 7-28 || “l773co8 | 4-%8 "|" lep6a02 | 18 
08245 | 522 || 934849 | joe || = ar7ae0e | 3-8 "226104 | 17 
708458 | 3'px || .934274 | jon ||| 774184 | 4°23 225816 | 16 
708670 | Sigg || 1984199 | J°6> || rama | 8 "225529 | 15 
108882 | 3°53 984123 | 4°98 WATS | 4 ng 1225241 | 14 
709001 | 3'r3 || 934048 | joe || 775046 | 4° "224954 | 13 
209806 | ‘sq || .983073 | joe *||: aves | 4-23 "924667 | 12 
709518 | 5"p3 || .933898 | j76> || .7zee | 4-3 "924379 | 11 
a sige) ||" 080822) joe: ||. srese08 | ae 224092 | 10 
9. 709941 9.933747 | 1, 9.776195 |, ng | 10.223805 | 9 
10158 | 3-P3'\) fe TTL crease | Ske | | aap | 8 
710364 | 359 197 TUGT68 | Ang 223232 | 7 
710375 | 3°59 93352 195 171055 | Ang 222945 | 6 
-710786 | 3°55 933445 |} o9 T7342 | om .222658 | 5 
-710997 | 3°55 933369 | 3 '9* TUE | Ging 222372 | 4 
ePM208 |. oo en 933293 | 46 a UUT915 497 222085 | 3 
711419 3 50 933217 | oy 78201 | 4'ng 221799 | 2 
.711629 | 350 933141 — . 778488 ae 221512 |} t 
9.711839 | 3-59 || 9 932066 | 1-25 || givzerza | 4-7 | 10/221226 | 0 
Cosine. ' D.1". Sine. Dyer” Cotang. | D. 1” Tang. nie 


COSINES, TANGENTS, AND COTANGENTS. 


aA 
Oo 
° 


Sine. Dri". Cosine. | D. 1”. 
9.711889 xo || 9.983066 a 
712050 | 3-35 || 92990 | 2.27 
seen): | | Sa 932914 | 1.37 
712469 | 3°55 932838 197 
piee7 |) 2 932762 | 1.27 
“719689 | 3-50 932685 | 1-38 
713098 | 3-58 982609 | 1-3¢ 
713308 | 3:88 992533 | 1-37 
earsatT | | oop 932457 | 1.37 
KhyAQKhO . < é 
-713935 | 3°48 veal 
9.714144 9.93222 . 
"714352 a "932151 es 
714561 | 3-48 932075 | 1-32 
714769 3 48 -931998 | "58 
mags | 3-48 -osi921 | 1.58 
Aaa. || 3.28 hilt Seeeeee (11 18s 
~é 5394. 3 47 931 4 68 1 98 
“715602 | 3-42 931691 | 1-38 
715809 | 3-45 931614 | 1-28 
M6017 | 3-46 931537 15 
9.716224 ~ || 9.981460 
716432 | 3-47 || ° “931383 i 
716639 | 3-48 931306 | 1-58 
“716846 | 3-43 931229 | 1-58 
a1708a | (3°03 931152 | 1-38 
tea |. |e 931075 | 4-58 
717466 | 3-4 980998 | 1-58 
gers |. 1303 930921 | 1-35 
717879 | 3-3 930843 | 1-58 
718085 | 3:43 930766 | 1-35 
9.718291 9.930688 
718497 | 3-48 || ‘osoe1i | 1-28 
“718703 | 3-43 coponas | 1-8 
718909 ie 930456 eo 
“M1911: "93037 
“719820 | 3-43 || 1930300 | 1-30 
719595 | 3.2 930228 | 1-55 
visza0 | 3-2 930145 | 1-3) 
prigees | 12-8 930067 | 2-3) 
720140 | 3-3 929989 | 3°39 
9.720845 9.929911 
720549 | 3-49 || 920833 | 1-3) 
mania | 3-48 929755 | 1-35 
720958 | 3-40 929677 | 4-35 
yoiiee |’ (3-2 929599 | 1°39 
721366 | 3-40 929521 | 1-39 
Siero | 12, 40sttl | Wee 
woes |\ oS 929207 | +35 
9.722885 9.929129 
7ee5es | 3-38 -|| ‘920050 | 1-38 
T2201 | 33-2 928972 | 1-35 
vezoo, | 3-38 928893 | 1-55 
(23197 3 3 38 . 928815 { 4 29 
me -t HQKHHe vw 
423400 | (3°38 928736 | 4°39 
‘ 23603 3 37 § 92865 ( f 329 
723805 | 3-37 928578 | 1-35 
724007 | 3-32 928499 | 1-35 
9.724210 | 3" 9.928420 | 1-9 
Cosine. Sine. D, I. 


HY Cotane. 


TABLE XXV.—LOGARITHMIC SINES, 


Tang. 


9.778774 
779060 
779346 
779632 
779918 
780203 
780489 
18077 
781060 
781346 
781631 

9.781916 
782201 
782486 
E271 
782056 
783341 
783626 
783910 
184195 
784479 

9.784764 
785048 
785832 
785616 
785900 
786184 
786468 
186752 
787036 
787319 

9.787603 
787886 
788170 
788453 
788736 
789019 
789302 
789585 
789868 
790151 


9.790434 

(90716 

. (90999 

791281 

791563 
~ .791846 
.792128 
792410 
. 792692 
19297 
. 193256 
. 193538 
. ($3819 
94101 
794883 
794664 
1949 16 
795227 
795508 
9.79578) 


© 


D, 1". 


| 
| 


mF BES ae eg ee tg gets 


NW OW NOC OTST ST OLOT-SIONt ISAT IAas 


IIIS 
C9 09 09 OD 


4.68 


Cotang. 


10.221226 
220940 
220654 
220368 
220082 
219797 
219511 
219225 
218940 
218654 
218369 

10.218084 
217799 
217514 
217229 
216944 
216659 
216374 
216090 
215805 
215521 


10.215236 
-214952 
214668 
214884 
-214100 
-213816 
213532 
213248 
212964 
212681 

10.2123897 
212114 
-211830 
.211547 
-211264 
. 210981 
210698 
.210415 
.2101382 
- 209849 


10. 209566 
. 209284 
209001 
208719 
. 2084387 
208154 
207872 
.207590 
207308 
207026 

10.206744 
206462 
.206181 
. 205899 
. 205617 
- 205336 
205054 
204773 
204492 

10. 204211 


Sine. 


9.724210 
724412 
124614 


724816 | 


725017 
.725219 
725420 
(25622 
T2582 
726024 
- (26225 
9.726426 
.126626 


730217 

| 9 730415 
“730613 
“730814 
"731009 
"731206 
"731404 
"731602 
"731799 
"731998 
732193 
| 9.732390 
9 | .732587 
3 | 739784 
| "1732980 
“733177 
"733373 
"733569 
733765 
"733961 
"734157 
9.734353 
734549 

| i734744 
"734939 
"735135 
"735830 


. 135525 


735719 | 
| .735914 
| 9.736109 


Cosine. 


COSINES, TANGENTS, AND COTANGENTS. 


Tang. 


Cotang. 


je) 
Se) 


ee 


oO 


WW WNWWWW2 ra ra IWweOwwe worwwowowwc Goo? 29 ¢ 9 09 C OC > oo co O9 09 OH CY OH 


os 0) 09 Co 
D2 Ww? 


9 928420 
928342 
928263 
928183 
“928104 
“928025 
“927946 | 
“927867 
927787 
“927708 
927629 
927549 
927470 
-927390 
927310 | 
(927231 
“927151 
927071 
926991 
926911 
926831 
926751 
926671 
926591 
926511 
926431 
926351 
926270 
926190 
-926110 
926029 
925949 
925868 
925788 
925707 
925626 
925545 
925465 
925384 
925303 


9252 


Todi 


925141 
925060 
924979 
924897 

924816 
924735 
924654 
. 924572 
. 924491 
. 924409 


924328 
924246 
924164 
. 924083 
924001 
9238919 
. 928837 


or 


.923759 | 
923673 


923591 


| 9.812517 


9.795789 
796070 
796351 
. 796632 
796913 
-T9T19A 
T9T4AT4 
19TTSS 
798036 
.798316 
798596 
198877 
799157 
799437 
st OOe17 
W99997 
800277 
800557 
800836 
.801116 
801896 
801675 
801955 
802234 
802518 
802792 
. 808072 
803351 
803630 
803909 
804187 


9.804466 
804745 
805023 
805302 | 
.805580 
805859 
.8061387 
806415 
806693 
806971 

9.807249 
807527 
807805 
808083 
808361 
.808538 
8038916 
.809193 
809471 
809748 

9.810025 
.810302 
810580 
810857 
811134 
.811410 
.811687 
811964 
812241 


is) 


ie) 


10.204211 

203930 | 
203649 | 5! 
203368 | 5 

. 203087 ) 
202806 
202526 
202245 
201964 
201684 
201404 
201123 
200843 
200563 
200283 
200003 
199723 
199443 
199164 
198884 
198604 
198325 
198045 
197766 
197487 
197208 
196928 
196649 | ; 
.196370 | 32 
.196091 | ; 

195813 


195584 
.195255 | % 
194977 
. 194698 
. 194420 
194141 
193863 
193585 
193307 
198029 
192751 
192473 
.192195 
.191917 
191639 
.191362 
191084 
190807 
190529 
190252 
189975 
189698 
. 189420 
189143 
188866 
188590 
188313 
. 188036 
187759 

10.187483 


| Cosine. | 


Sine. 


Cotang. | 


Tang. 


o 


09 He OT =F COW 


~ 


Ord 


~ 


~ 


10 


SOANQIDUR WMHS 


TABLE XXV.—LOGARITHMIC SINES, 


Sine. Cosine. Tang. Cotang. | 
ie | 
736109 9.923591 9.812517 87483 | 
736303 "923509 “Siar | 4-620] TORR 
736498 "993497 ‘813070 | 4-6 186930 | 
136692 923345 "813347 | 4-6 "186653. | 
736886 923263 "813623 | 4-8 "486377 | 
737080 (923181 “813899 | 4:6 136101 | 
37244 923098 “814176 | 4-6 "185824 | 
7346 "923016 'gi4452 | 4-6 "185548 | 5 
731661 922983 814728 | 4-8 “js53r2 
“T3785 922851 815004 ‘4 “TR008 | 
ss he 922768 815280 | 7 184720 | 
1382 922686 815555 5 
"7B8434 "929603 "815831 | 4:6 ae his | 
738627 "922590 ‘816107 | +: "183893 | 
738820 922438 916882 | 4-5 "183618 
739013 "929355 “816658 | 4-9 483342 
“7739206 "929972 "816983 | 4-5 483067 | 
739308 922189 | "917209 | 4: "182791 
(39590 922106 ‘gi7484 | 4: "182516 | 
739783 922023 ‘g17759 | 4:5 182241 
139015 921940 818035 | 4° 181965: | 
9.740167 9.921857 818310 -¢ | 10.181690 | ; 
740850 | 3- 92177 "g18585 | 4:4 181415 | 
740550 | 3. "921601 818860 | 4: "181140 | 
140742 | 3: 921607 (819135 | 4: "180865 | 
740984 | 3. “921524 ‘819410 | 4: 180590 
iatie5 | 3. 921441 ‘819684 | 4: "180316 | 
741316 | 3: 921357 819959 | 4+ 180041 
741508 | 3: “921274 820234 | 4 179766 
741699 | 3° “991190 "820508 | 4+ 179492 
741889 | 3-4 921107 820783 = 179217 
9.742080] 5 9.921023 9821057 10, 178943 
ar |B "920989 | 921332 | 4: "178668 
742 mage | , 3: "920856 821606 | 4: 178394 
742652 | 3. "920772 "g21880 | 4: 478120 
74e842 | 3: "920688 320154 | 4: 177846 
743083 (3. 920604 822429 | 4. ATT 
743223 | 3. "920520 “822708 4. 97297 
743413 | 3-4 920436 ‘go0977 | 4: 177023 
43602 | 3-7 "920352 “gazes, | 4! 176749 
TABI | 3 "920268 823504 ; "176476 
743982 | 9 4 920184 9 823798 ; 10.176202 
mai | 3-5 "920099 | 1824072 | 4:2 "175928 
744361 | 3-1 920015 824345 | 4 175655 
744550 | 3-42 || 919931 | 824619 | 4: “175881 
744739 | 3° 919846 "824803 | 4 "15107 
744928 | 3-1, 919762 925166 | 4: 174834 
TABI? | 374 919677 825439 | 4: "174561 
745806 | 3-4 "919593 "925713.| 4: "174287 
T45494 | 8: “919508 “925986 | 4:5 T4014 
745683 | 3- 919424 26259 | 4:55 1737 
45083 | 49 26259 | 4-22 “173741 
T4871 | 9 4x || 9.919339 826582 | 4 xx | 10.173468 | 
746060 3°13 919254 “926805 | 4:59 "173195 | 
746248 | 3.13 || ‘oro169 “27078 | 4:25 |. “172922 | 
746486 | 3-18 “919085 827351 | 4:58 "472649 
746624 | 3:43 || “919000 827624 | 455° | ' Ci7a3v6 
796812 | 3-13 || “orgots 27807 | 4:25 172108 | 
746999 | 3-12 || 918830 828170 | 4-55 |” 7171880 
TAS; | 3-18 “918745 828449 | 4-53 171558 | 
“47874 | 3745 "918659 “gos715 | 4:99 “T1285 
9.747562 | 3-13. |) 9 oresey | gsosoe7 | 4°23 | y07 174023 | 
Cosine, | D. 1’. Sine. -Cotang, er DA Be Tang. 


a 


CosI 
NES, TANGENTS, AND COTANGENTS 


/ ° 
Sine 
. D. Cosine. | D. 1" | Ta; 
—- | ng. D. i"; }-Cotang. | 7 
4 | 9.747562 | 9 yo. || 9.918574 | 
TATT49 SfO® |My ced | | = eee ees 
2 | narperes = a 9 918489 1.42 | 9.828987 4 
BY tater Ser i) Nereis | 1-2 929260 | 4-23 ee ee 
4{ -v4gsi0 | 3-12 meieta (Lek! Mi Mpanene i eaten | ee 
SB haaeeene |S aida Ih aeptere | 1142 || -829805 ao \T ote unige | Bs 
G6 | :7486s3 | 3-10 ‘918147 | 1-48 Be, ae ee Uden | bs 
7| ‘ragg70 | 3-22 ‘918062 | 1-42 -830349 4.53 ease | 56 
8 | 749056 | 3-10 ‘917976 | 1-48 “g30621 | 4°23 Pe eeme 
Bh teas | Saba? Ni ceteane 14s || -830803 | 4783 ere, | Be 
10 | {749429 | 3-10 oir | 1-43. |] ceatase ae ieeeas | Be 
a “y f°) 8.40 ‘gizzi9 | 1-43 831437 | {PS - 168685. | 02 
11 | 9.749615 1.42 "831709 | 4:22 .168563 | 51 
12 -T49801 3.10 9.917634 . : 4.53 .168291 | 50 
Bot Shager | 8.10 |l venues ifdat || Seer hs 4 eeeeaie (eds 
14} “750{72 | 3-08 ‘g17462 | 1-43 882253 | 7° 53 peat 49 
Me Bicpees 1 Sal0* Ill cae tag || 892525 | 4iee | ee | a 
16 150543 3.08 917290 1.43 8327 96 4: = 167 ’, ! 
Bul Gress || S10) 1) Gama 1:43 || 838068 | 4-25 Mees | ae 
i8| 750914 | 3-08 ‘oi7tig | 1-43 833339) oe byt | 3 
19 | 751099 | 3-08 ‘917032 | 1-43 eooe! Re oe a 
90 | “751284 | 3-98 ‘916946 | 1-43 933882 | “4-33 1664 sg Be 
21 | 9.751469 ; 1.43 "ga44on | 4.52 165846 | 41 
| TOI! | 3 0p Maier | 1a |ll See Puc nee | 
wm | 751054 | Bog |] 916687 | tgp || 83404 
ae aes 3.08 "|| “o1e800 | 1-45 834967 | 7b 1 ere | ae 
Be serach | | 8508" ||| Seteer 143 || “S828 | Gop ees | ee 
26. | “752392 | 3-00 "916497 | 1-4 "885509 | 4°R6 ei | oe 
97 | l75e5t6 | 3-0¢ ‘916341 | 1-43 - 835780 yo teres oe 
el Warsece (18M! |) Satetey 145 || 886051 | 4-35 seaniy ‘LSee 
99 | ‘sa944 | 3-07 ‘916167 | 1-49 836322 4.52 % Sees | ea 
3) | ‘ssajog | 8-07 ‘g16081 | 1-43 836593 | 4:25 pete 33 
. 23 | 3 'or “915994 | 1-45 "g36a64 | 4:52 .163407 | 382 
31 | 9.753312 1.45 "e37q34 | 4.90 .163136 | 31 
39 | "753195 | 3-99 9.915907 ue | 4.52 "162866 | 30 
33 753679 3.07 . 915820 1.45 9.837405 | 4.5 10.162595 | 2 
S| 753862 3.07 || ante aaa tee Rage iid Bite 162325 | 2 
35.1 95 aT} 91564 45 001920 peed aaRe aoe 
Onn ages | | 3.05 915648 | G5 || 838210 | bp 162054 | 27 
7 |. 754412 | 3-0 “ieire | 1-45 |] “Shersy 4.52 | 1161518 | a 
38 ws4ng5 | 8-05 |] "915385 | 1-45 -83875% oO "46 ; 3 1 
39 | RAT > ||, Zorpeo7 | 1-47 939027 | 4:20 -161243 | 24 
to | 751060 303 || peewee 1H 1 aS 1) peu 4i608. | cigars | ge 
Suen (i) comes [i age 930568 | 4:52 -160703 | 22 
41 | 9.755145 : 1:47 "g3ggag | 4-90 160432 | 21 
42 | 755326 | 3-9 || 9.915035 4.50 .160162 | 20 
43.| “755508 | 3-93 || "914948 | 1-45 9.840108 | 4 5 | 10.159892 
ts | -ra5om0 | 33 914860 | 445 810873} 45) | «150622 19 
5 | ‘yssgr2 | 3-03 || (ee eRe ¢ 409; Aiag "159852 
46 | 756054 |. 3-08 "914685 | 1-46 4017 ee aoaae | de 
Bit adeee bi 8208? ||| agraeee tag ||) cBatte7 | Diape | pa ede 
43.| ‘75641g | 3-03 ‘914510 | 1-46 “841457 | 4-2) eee ae 
49 | 756600 | 3-03 ‘914492 | 1-42 SALT2¢ oe oes Bs 
Bel Beets | 1308 ||| Sopris Loeyt |i) peaaeoe era ean de 
82 | 3'o9 || «914246 1.47 842266 4.50 -158004 | 12 
51 | 9.756963 ant 1.47 "gy9535 | 4-48 157734 | 11 
53 957326 3 03 .914070 1.47 -¢ 42805 78 10.157195 
Br iameere f82 * ||| capeeas 147 i pions | 448 | 156: aie 
55 | 757688 | 3-02 ‘gizso4 | 1-47 949313 | 4.48 fee 8 
56 | “757869 | 3-02 "913806 | 1-44 813612 4.48 efuee sf 
57 | 7sgos0 | 3-02 ‘913718 | 1-42 gt3eee | 4.38 eon 6 
58 | 758290 | 3-00 ‘913630 | 1-40 | 844151 ‘oe perce 5 
59 | 2758411 | 3-02 |, ores | IMB Il! sages fue Ween ag 
ed cde ok. (8.00: || ‘oigtaass a || 844689 | 24g Rey ey 
ee  oapegeral |. 150 * It eines 4.38 55811 | 2 
7 | Cosi ; meres || g g45an7 | 4:48 ..155042 | 1 
aceak D. 1 Sine eo ee C | 10.154773 | O 
; Joi. ota S= 
ang. | Dek: Tang. 


TABLE XXV.—LOGARITHMIC SINES, 


r | 
Ab Sine. ih Del" ett!) Casime: 1.8") 1 Tang. | D.1". | Cotang. | / 
| pee =< ie 
3365 520% | 10.154773 | 60 
0 | 9.758591 | 5 gp || 9.913365 | 1 4g || 9.45201 aes 10. 154773 | 6 
| eypegse | 8:00. ||. Seaeene |: 1ea8s |) Teateeey |e olay) | eee ee 
S| Swsgice | 8006 || Soteney | tel cl Pagers (dpi 9 dee 
3 | 759182 | 3-00 913099 | 3-48 816033 | 948 153067 | Br 
B | coepadon | (82000 |] pigs | tear ll Babee? | aide || ge eee 
8| “tore | 3-00 || “Oioegs | 1-48 |] Bienen | 4.48 | tBRIR0 | Be 
7 | cesasse | 2-00 |] “orerat | 1-48 |] Ramos | 4:48 | 1e8np0 | Bs 
b | Sreopet || 2e98-||) Serages | 1eaB |): Geatee? | Gaede Ses 
6 | Sopeoat (8-00 ||) SBipeee |. aBy ||| Beet Tia ay ell) aameeee ae 
3 | “heosso | 298+|| coir | 1aB ct Seyeee | 4aeo| | Rete 
10 760390 2.98 912477 1.48 . ( 4,47 | i i e 
Vet t os hE 8 
11 | 9.760569 } 9 9g || 9.912388 | 4g 9.818181 gel ae 151819 49 
12} -.760748 | 3‘o9 912209 | 4°48 S189 |p “151551 | 48 
14 | 761106 | $38 912121 | 185 848086 | fr 151014 | 46 
ie | “4e1ie4 | 2-98 ‘pitode | 1-48 "849502 ae 150478 | 44 
of 5 ) 2 97 2 e a 1.48 Ts ed ‘ "150210 8B 
18 | ‘reir | 2-98 || “utes | 1.00 "850057 re 149943 | 43 
| Pf 4 (9) 97 2d ioe 1.48 . xd ; ned ; iy . ; ar 
19 | .761999 et -o11674 | +58 850325 2 149675 at 
21 | 9.762356 | 9 97 || 9.911495 | 4 59 9.850861 que | Gs 149139 39 
ba |. ree5a4 | | 2:87 911405 | 1-20 S51129 | 4-45 “14881 | 38 
3 Eee 2.95 “pitase (18 "851664 rp "148336 | 36 
- (06000 ¢ Y . we 1.50 eae A , 3. 
| 25 | .763067 4 dl 911186 | 3-39 81931 ae 148069 35 
26: |" 2768045 | S:af 911046 | 1-39 852199 | 445 147901 | 3 
7 | -68422 | 9" oe -910956 | 4 "59 Poemee |! 4d Soe lee 
i 28 | 763600 | 3-94 910866 | 3:20 852133 | 7 141267 | 82 
29 163777 2 95 91077 : 1 50 eos 38 4.45 146732 30 
30 . 763954 9 ‘ 95 . 910686 1 7 50 . 85826 4 § 45 ° rf a 
. =e =4 35 ahd 46! € 
31 | 9.764131 | 9 ox || 9.910596 | 1 '., 9.853535 dasa 40465 29 
a2 | .764308 | §:22 910508 | 3-78 858802 | 445 146198 | 28 
33 | . 764485 | 3-02 o104i5 | 1-8 854069 | 4-42 15081 | 
I 764662 igy 910325 2 854336 > 145664 | 26 
34) .764662 | 9°93 terre |i Bey Sorin es ore hae 
35 | .7e4sa8 | 5:08 910085 || .1+9) 854003 | 445 145897 | 2 
Be HBetgH 2.98 BOREL 1.50 "855137 re "144863 | 9: 
3” T6515 3 9: 91005 159 -855 a ate : 
38 | 765867 | 2:98 909963 | T"pe $5408 aa “144596 2 
89] . 765544 | 9°98 -909873 | 4759 eer | 04 ia ie. 
40 | .765720| 5-33 909782 | -}:D5 855938 | 4-23 1410 
Hil 41 | 9.765896 | 9 93 || 9.909601 | 45 9.856204 ye 10. 148796 19 
qa 42] .766072 | 9°99 909601 | 3°55 856471 | 4°43 ie lee 
aaa As 766247 909510 ‘ 856737 i 143265 7 
A ri Lives 2.93 909419 | 1-52 857004 hoe 142996 | 16 
i caer. 1o TG Byes 2 1.52 Saath 4 149730 | 15 
wa) 45 | 766598 | 3:92 909828 | 1 °p8 857270 | 45 142730 | 15 
46) .(66774 | 999 909237 | 459 ete | dake reteal es. 
47 | 766049 | 5-05 909146 | 1°28 851803 | 443 142197 | 13 
49 | .767300 | 9'99 -908964 | 3 "59 Seas | 4s alana 1a 
50 ° T67475 9 } 90 . 908873 1 ; 53 O00 i 4 : 43 . ae 
51 | 9.767649 | o9 || 9.908781 | 4 ‘5 9.858808 “shy 10. 141122 9 
ef ihatenn’ | eee | dese 1.58 360400 4.43 "140600 | % 
53 767998 290 -t 00) 1.53 . 2 = 4.438 1 10334 6 
Bd |. 208173 |~/3 80 908507 | 5 "p3 859608 | 7°43 40334 | 6 
55 | 768348 | 2-9 908416 |} ps “850932 | 4-48 140068 | 5 
56 || q7esne2 | 2:20 908324 Bes 800198 ie ‘ 139802 4 
Bz | 768607 | 3-92 9082! es 860464 | 4-43 139536 | 3 
58 | .7esevt | 2 60 908141 | 7:23 860730 | 4-43 139210 | 2 
59 | 760045 | 2: ‘gos049 | 1-5 860995 139005 | 1 
60 | grees | 2.90 9.907958 | 1-52 |! 9'sero6r | 4:43 .| 407438739 | 0 
| / | Cosine. | D. 1”. Sine. | D. 1". Cotang. | D. 1" Tang. : 


| 
, Sine. 
| 
| 0 | 9.769219 
Bet 769393 
2 . 769566 
3 769740 
Gav! 769913 
PD |.> 770087 
6 | .770260 
| @ | .770433 
| 8 | .770606 
9 T0779 
10 770952 
11 | 9.771125 
2 | .%'71298 
13 | .771470 
14 | .771643 
15 | .771815 
16 |° .771987 
17 772159 
18 772331 
19 |! 7725038 
2 ~ 772675 
21 | 9.772847 
22 . 773018 
23 (73190 
24 .773861 
25 STLOOOD 
26 773704 
27 ~TI8875 
28 774046 
29 Ne42de 
30 .7743888 
; 31] 9.774558 
| 82 | 774729 
33 | .774899 
34 775070 
35 ~ 175240 
36 . 775410 
30 775580 
38 TIS T50 
39 775920 
40 .776090 
41 | 9.776259 
42 716429 
43 |  .776598 
44 776768 
45 776937 
46 ~777106 
47 UTD 
48 1071444 
49 77613 
50 T7781 
51 | 9.777950 
52 | .778119 
BS cane? 
54 78455 
DD | 778624 
56 778792 
57 | 778960 
| 58} .779128 
59 779295 
60 779463 


| 
Ds 


cM) 


|’ 1 Cosine. 


o 


Wwrwrmwrwwwwwn wnwnwnwnnwnnwnwnnwwna 


BW W WO WW W WWW WWW WWW W WWW WWW WWNWWNWW)DW 


8 


MAE AQVOU SPST MEI AEE ak 


Cosine. 


9.907958 
| 907866 
IOVT7T4 
. 907682 
. 907590 
1}. 907498 
|| .907406 
.907314 
907222 
907129 
907037 
9.906945 
. 906852 
. 906760 
. 906667 
. 906575 
. 906482 
. 906389 
. 906296 
. 906204 
.906111 


9.906018 

905925 
|| 905832 
|| .905739 
|| 905645 
|| 905552 

905459 

905366 

905272 

905179 
9.905085 
|| .904992 
904898 
904804 
904711 
904617 
904523 
904429 
904335 
(904241 


9.904147 
904053 
903959 
903864 
90377 
. 903676 
903581 

i| .903487 

| 903392 

|| .908298 

| 9.903203 

| .903108 

‘| 903014. 

| ,902919 

902824 

i} ,902729 

l| . 902634 

|| 902589 

| 902444 

|| 9.902349 


| Sine. 


Dei", 


BB | 
BB || 


ie} 


Pek eh fre fh fee fk fed fe fk fe fed fame fk peck eh fr free foemch fore fre fame fore fom prc feel fem fmeh feed foe fom farm bere feel fremh prem peak fom fem fem fom prem fre fem fmm from fom free from fommh meh. fem fom famh fem fem free fed feed peek fed 
Pe Nc Co ati, ae ED CL ers ee ek re SOS ATMO ile Sle tic. OLE ett ee te Cite Sa eel ee Na OM te ate yan ea Me Dare Aire) he 
On 
NX 


Tang. 


| 9.861261 


.861527 
861792 
862058 
862323 
862589 
862854 
.863119 
863385 
863650 
863915 


9.864180 
864445 
.864710 
.864975 
865240 
865505 
86577 
866035 
866300 
866564 

9.866829 
867094 
867358 
867623 
867887 
868152 
868416 
.868680 
868945 
869209 


9.869473 
.869737 
870001 
870265 
870529 
870793 
871057 
871821 
871585 
871849 


.872112 
872376 
.872640 
.872903 
.873167 
.873430 
. 873694 
. 873957 
. 874220 
874484 
9.874747 
.875010 
.875273 
SOLOD3 « 
.875800 

. 876063 
876326 
.876589 
876852 
9.877114 


Cotang. 


COSINES, TANGENTS, AND COTANGENTS. 


D, 1". 


HSS 


SMW WWOWWWW 


nC 


PEP LEB 
OOOCWwonwow 


He 
Ps 


4,40 


geo, Seid ma oo ian - g 
io.2) 


Cotang. 


10.138739 
.138473 
. 188208 
.187942 
.137677 
137411 
.137146 
.136881 
.136615 
. 136350 
.136085 

10. 135820 
. 135555 
. 135290 
.135025 
. 134760 
, 134495 
134230 
. 133965 
. 133700 
.183486 

10.13317 
. 132906 
. 182642 
. 182377 
. 182113 
. 1381848 
.181584 
. 131320 
.131055 
.180791 

10.130527 
. 1302638 
. 129999 
.129%85 
129471 
.129207 
. 128943 
. 128679 
.128415 
128151 


10.127888 
127624 
127360 
127097. 


126833 


126570 
126306 


"426043 
"125780 


. 125516 


10. 125253 


. 124990 | 
£124727 | 
124463 | 
124200 


1238937 
123674 
123411 
123148 
10. 122886 


TNIGS 


Cc 


Ort Cor ot 


~ 


TABLE XXV.—LOGARITHMIC SINES, 


Det Cosine: | D, 1’. Tang. D. 1”. | Cotang. 


9.877114 
| Oat? 
.877640 
877603 
.878165 
. 878428 
.878691 
.878953 
,879216 
.879478 
879741 
9.880003 
880265 
. 880528 
.880790 
.881052 
.881314 
881577 
. 881839 
. 882101 
.882863 
9.882625 
. 882887 
.883148 
883410 
. 883672 
883934 
.884196 
. 884457 
. 884719 
.884980 
9.885242 
. 885504 
. 885765 
886026 
.886288 
. 886549 
886811 
.887072 
. 887333 
887594 
9.887855 
. 888116 
. 888378 
.888639 
888900 
. 889161 
. 889421 
. 859682 
.889943 
. 890204 


9.890465 
890725 | 
.890986 
.891247 
891507 
891768 
892028 
. 892289 

| ,892549 
9.892810 


Cotang. | D. 1’. Tang. 


10122886 
122623 
122360 
122097 
.121835 
121572 
121809 
121047 
120784 
. 120522 
.120259 
119997 
119735 
119472 
119210 
118948 
118686 
118423 
.118%61 
117899 
117637 


117875 
117118 
116852 
. 116590 
116328 
116066 
. 115804 
115548 
115281 
115020 


114758 
.114496 
114235 
118974 
1138712 
118451 
118189 
112928 
112667 
112406 
10.112145 
111884 
.111622 
111861 
-111100 
110839 | 
110579 
.110318 
-110057 
. 109796 


109585 
.109275 | 
.109014 | 
. 108753 
.108493 | 
. 108232 
107972 | 
107711 
.107451 
10.107190 


779463 9.902349 
479681 | <*5 902253 
W79798 | 5° 902158 
7799668 | 5" 902063 
780133 | 574 901967 
780300 | 5°4 901872 
780467 | 5° 901776 
780634 | 5° 901681 
730801 | 4" 901585 
780968 | 5° 901490 
781134 | 5° 901394 
781301 we || 9.901298 
781468 | 5° 901202 
781634 | 5° 901106 
781800 | 5° 901010 
781966 900914 
7182132 900818 
782298 900722 
782464 900626 
782630 900529 
782796 900433 
782961 900337 
783127 900240 
783292 900144 
783458 900047 
783623 899951 
783788 899854 
783953 899757 
784118 899660 
784282 899564 
784447 899467 
784612 9.899370 
784776 899273 
784941 .899176 
785105 899078 
785269 898981 
785433 898884 
785597 898787 
785761 898689 
785925 898592 
786089 898494 


786252 9.898397 
786416 898299 
786579 898202 
786742 898104 
786906 898006 
787069 897908 
787232 897810 
187395 897712 
BUST 897614 
787720 897516 
487883 9.897418 
188045 897320 
7838208 897222 
788370 897123 
788532 | of 897025 
788694 | =f 896926 
788856 | 896828 
789018 | >°f 896729 
789180 | 5-f 896631 
| 9.789342|  *° 9.896532 


9 G2 0° 


BSMISU FL COWH SO | 


BS EAP a ee eg gg + 


WW CWC OCI OD OTD GLOTRS COTOTOTOTR OI OTS NEN 


Si dnt Boas Jn Teas 


WwwWwwoOWwte CWO WWW CO CU CU wt tt) CHC a eg 


ESSA! Ov Ov OT OU 09 OU OF OU 3 OF RRRRSRSRES POTS OU SETS OT SSE SE AE AAI AE = HAS BO 3 CO =2 0 GOOD 


Fare Se Sa 


SWS VWOCWNNVWWWWS 


~ 7 


WRPWWWwWwWwww wwwwwwwwww 


Cosine. F Sine. 


oy eT RAE ee ee Cte test y. >> 2S > > 2S > SD ie ee i 


COSINES, TANGENTS, AND COTANGENTS, 


wo © 
= © 


ivy) 
iva 


~ 
i) 


Jeo) 


Le 


/ 
0 
14 
2 
3 
4 | 
5 
6 | 
ii 
8 
9 
10 | 
11 | 
oe 
13 | 
14 | 
15 | 
16 | 
q 
18 
19 
22 
23 
24 | 
25 
26 


| 
60 | 


Ae 


Sine. 


9.789342 
789504 
789665 
- 789827 
789988 
.790149 
. 790310 
-790471 
. 790632 
790793 
- 790954 


9.791115 
791275 
.791436 
~ 791596 
T9157 
-T91917 
192077 
792237 
- 792397 
- 192557 
. 792716 
792876 
.793035 
.793195 
793354 
. 798514 
. 793673 
798832 
.793991 
- 794150 


794308 
(94467 
794626 
794784 
(94942 
795101 
- €95259 
95417 
95575 


195733 


ve} 


| 9.795891 


. 796049 
796206 


. 796364 


796521 
796679 
796836 
796993 


797150 


797307 | 
| 9.797464 


797621 
GIG 


TIT 9B4 


.798091 
798247 


798403 
. 798560 
. 798716 
9.798872 


Cosine. 


2. 


TWWWWWD 


gS 1920} 


IWWNWNWWWWWW 


G9 WD 09 WD LO 


WW WWW WWW 


WWW NWWWWWW WW WWW WWW WW 


65 


| 
| 
| 
| 


Cosine. 


|| 9.896532 


. 896433 
.896335 
. 896236 
.896137 
.896038 
.895939 
.895840 
.895741 
.895641 
.895542 


9.895443 
.895343 
. 895244 
.895145 
.895045 
.894945 
.894846 
894746 
.894646 
894546 
9.894446 
.894346 
.894246 
.894146 
.894046 
.893946 
.893846 
.893745 
.893645 
. 893544 


9.893444 
.893343 
-893243 
.893142 
893041 
.892940 
892839 
.892739 
.892638 
.892536 

9.892435 
. 892334 
892233 
.892132 
892030 
.891929 
891827 
891726 
891624 
891523 


9.891421 


.891319 


.891217 
.891115 
.891013 
.890911 


.890809 
. 890707 
.890605 


9.890503 | 


Sine. 


1);,.1?. 


Be ek ek ek ek ek ek el ek el ee pe ee pe he ep Re eh ee ee ee ee ek ee ee ep 
fer) 
2 


o 


Cotang. | / | 


141° 


Tang. De ehz, 
| 9.892810 | 4 9, | 10.107190 | 60 
893070 | 4°33 "106930 | 59 
.893331 4.33 .106669 | 58 
893501 | 4°33 106409 | 57 
mall ae eee 
yn bape 4.35 espe 
894372 | 4-23 "405628 | 54 
804632 | 4°33 "105368 | 53 
g4gg2 | 4-3 5 2 
Oe is | dees 
-895152 | 4°33 -104848 | 
.895412 4.33 .104588 | 50 
9.895072 | 4 95 | 10.104388 | 49 
895932 | 4°33 104068 | 48 
eopien | 38 | aatisese lugs 
"go67i2 | 4:33 "103288 | 45 
‘so6o71 | 4-32 "403029 | 44 
‘g97231 | 4-33 402769 | 43 
"gorag1 | 4-33 402509 | 4: 
ve Nake 4.33 Bl 2009 42 
S| 18 | ea 
-89E 4.33 -1019¢ 
9.898270 | 4 9 | 10.101730 | 39 
898530 | 4°35 101470 | 38 
aes a 
} 4 32 3 5 36 
899308 | 7°38 "100692 | 35 
899568 | 4°33 "400432 | 34 
899827 | 4°38 "400173 | 33 
900087 | 4°33 "099913 | 32 
900346 | 4°38 099654 | 31 
900605 | 4°35 "099395 | 30 
9.900864 | 4 9, | 10.099186 | 29 
901124 | 4°33 "098876 | 28 
901383 | 4°35 098617 | 27 
901642 | 4739 098358 | 26 
901901 | 4°35 “098099 | 25 
902160 | 4°3 097840 | 24 
902420 | 4°35 097580 | 28 
fon | 13 | wan |e 
Porcgeg wer ere Tae 
903197 4 39 .096803 290 
9.903456 | 4 9, | 10.096544 | 19 
OOSTI4 |: ae 096286 | 18 
“goiese | 4-82 | cogszes | 4g 
“go4491 | 4:32 "095509 | 15 
‘904750 | 4:32 “095250 | 14 
“905008 | 4-30 "091992 | 4: 
"905267 | 4:32 “094733 | 13 
905526 | 4-32 “o0ss74 | 41 
“905785 | 4:82 DOS Ss 
ot D185 4.30 094215 10 
9.906043 > | 10.093957 | 9 
906302 | 4-32 | ““‘o93698 | 8 
906560 va 093440 | 7 
emer rik ape eenea Wine 
ett yee O92923 | 5 
. 907336 4 ps .092664 4 
tm ( oes QI406 § 
Seog | 48 | ee 3 
908111 | 4°29 | 091889 | 4 
9.908369 | 4:39 | 40091631 | 0 
Cotang. | D. 1". Tang. Z 


— 
POUR WWEH COMMA RPWWOH SC 


Sine. 


9.798872 
- 799028 
199184 
. 799339 
- 799495 
799651 
- (99806 
799962 
800117 
-800272 
800427 


9.800582 
800737 
800892 
801047 
-801201 
801356 
-801511 
801665 
-801819 
.801973 

9.802128 
802282 
802436 
802589 
802748 
802897 
. 803050 
803204 
. 803357 
803511 


9.803664 
.803817 
.803970 
804123 
804276 
804428 
804581 
804734 
-804886 
805039 


9.805191 
-805343 
805495 
805647 
805799 
-805951 
.806103 
806254 
.806406 
806557 
.806709 
.806860 
807011 
807163 
807314 
807465 
807615 
807766 

807917 


| 9.808067 


Cosine. 


TABLE XXV,—LOGARITHMIC SINES, 


Or OF OF OF Or Or 
CO =2 2 =F CO CO “2 


~Ovrorgorororororor O14 


OUR OURS OT 3 2 OL 43 23 


WWWWNWWNWWWW WWW WW WW YW? 


B® 0 
a cron or: 


ris) 


5% WW 2 WWWWYD 


DW We 


ris) 


22929 


WWNwWWW 
Creer er ore 


“Delt 


Cosine. 


9.890503 
.890400 
.890298 
.890195 

- ,890093 
.889990 
.889888 
.889785 
.889682 
.889579 
.889477 
.889374 
.889271 
.889168 
889064 
.888961 
.888858 
. 888755 
.888651 
.888548 
.888444 


9.888341 
888237 
888134 
888030 
887926 
887822 
887718 
887614 
887510 


887406 © 


9.887302 
887198 
887093 
886989 
886885 
886780 
886676 
.886571 
.886466 
886362 

9.886257 
886152 
886047 
885942 
885837 
885732 
.885627 
885522 
885416 
885311 


9.885205 | 
885100 | 
.884994. 
.884889 
.884783 | 
.884677 


884572 


884466 


.884360 
9.884254 


Sine. 


s2orsze 


PIF I-73 


Beh ee ee ee eR pp Ph ek ek ek et ek pe ek ek ee ee ep ee pp 
z 2-6 ° ’ z ay 25 a Mtis ‘ 7 om os x x . + 2 
AB 3 3 OF 3-3 I-35 OF e 5 ah Oe Ra G as : - + 


3+ 53 a I+ 


| 
| 


9 


Tang. 


| 9.908369 


| 


. 908628 
. 908886 
. 909144 
. 909402 
. 909660 
.909918 
910177 
910435 


-916693 


-910951 


9.911209 
. 911467 
911725 
911982 
912240 
912498 
912756 
. 913014 
918271 
913529 


9.913787 
914044 
.914302 
.914560 
914817 
.915075 
915332 
915590 
915847 
.916104 


9.916362 
. 916619 
-916877 
. 917134 
.917391 
.917648 
. 917906 
. 918163 
. 918420 
-918677 


9.918934 
-919191 
.919448 
-919705 
. 919962 
. 920219 
. 920476 
. 920733 
.920990 
. 921247 
921503 
. 921760 
. 922017 
922274 
. 922530 
922787 
. 923044 
. 923300 
923557 

9.923814 


Dp... 


oo oo 
SSS38 


oo 0 O89 CO GO Ww Ww 
conan ask a) 


SSRRS 


do w wird 
DHDD 


Cotang. | D. 1”. 


WwWwnd Wwnwwwwnwww 


Cotang. 


10.091631 


- 090856 


-090598 


.090340 
-090082 
-089823 
089565 
.089307 


089049 


10.088791 
-088533 


088275 | 


.088018 
.087760 
087502 
087244 


086986 


086729 
086471 
10.086213 
085956 
.085698 
.085440 
085183 
084925 
-084668 
084410 
.084153 
083896 


10.083638 
083381 
0838123 
082866 
082609 
082352 
082094 


-081837 


.081580 
081323 
10.081066 
080809 
.080552 
080295 
.080038 
079781 


.079524 

079267 | 

.079010 

078753 * | 
10.078497 

. 78240 } 


077983 
077726 
017470 
077213 
076956 
076700 
076443 
10.076186 


Tang. 


| 60 
091372 | 
“091114 


COSINES, TANGENTS, AND COTANGENTS. 


" | Sine. | D. 1°. | Cosine. | D. 1’. 
| 
0 | 9.808067 | 2 59 } 9.884254 | Ly 
Ba aeyeoes k 20d: ll eresasen |: Lc 
3} isossi9 | 2-2 || ‘aesgag | 1-77 
4 | “gogg69 | 2-29 || “ggagog | 1-78 
| ~OUC 69 5) 50 | 883829 1 v7 
SS ecrees il BBO | gh 
Elicia | 20d dives tie 
| amo: | sp | ee a 
sai ~OUe y OF f Oe) ais 
} : 92 2.50 | SOK B ni 
S| -amost | 30 |) SO] TR 
CULE eos 2.50 Taek £27 
11 | 9.809718 | 9 59 || 9.883084 17 
2| .809868 | $"4g 882977 | yay 
‘ > Pied aw. OQr’ 4 
| eur | a0 | eet | 
15 | 1810316 wid "882657 ie 
> ’ v4 We RRO } 4 
17 | isioera | 2-48 || cgqias | 17 
18.| ‘siov63 | 2-48 "982336 | 1-78 
19 | Sioa | 343 |] 882229) 16) 
20 | .811061 | 9'48 882121 | 5"p 
21 | 9.811210 | 9 yo |] 9.882014 | | » 
2 | > 811358 | 3 74o -881907'| +745 
23 | ..811507 | "jo 881799 | 5 "p 
OA 2 << BOO . 
| eine 2 48 -881692 1.80 
26 | .811952 | 9 47 881407 | "a9 
S| oan) ae | Bae) is 
29 | (812396 | 2-44 || “geri53 | 1-80 
30 | .gi2544 | 2-49 || ‘agiogg | 1-48 
upp 2.47 . 1.80 
31 | 9.812692 | 5 yw || 9.880938 
aa | 812810 | 347 || 880880 | 1p 
33 | .812988 | 9°45 880722 | 3 "65 
36 | ceisis0 | 2-45 || “Bogaye | 1:80 
me Ol wagers 224% “agqado |’ 1-80 
peli cameos |; 24m {| prgepecs || 1:82 
Bel betters, |) 2240p || or ORO") 7280 
39 . 813872 | 9 45 -880072 1 82 
40 .814019 | ° 45 .879963 1 80 
41 | 9.814166 | , || 9.879855 
¢ n O46 2.45 | LOK AR 1.82 
42 ing |! Sap | pie 1.82 
pul Roaeeacr |) Qedee- |) peateee | 7 Be 
44 | 814607 } >) 43 ~ 879529 1 go 
45 | .814753 | 2 45 -879420 | 5 "g5 
Ak fe lot . 
| som) oa || aa | TB 
48 .815193 | 343 || ‘gr9093 | 1-82 
49 | .815339 | 9°43 878984 | 5°65 
50 .815485 2.45 878875 1.82 
51 | 9.815632 | 9 43 || 9.878766 | 1 go 
Bel dicitaes || 48> It Gece |. one 
Bi | “sioc9 | 2-42 |) ‘Sram | 1-8 
55 | .816215 | S77. .878328 | 3:3 
56 | 1816361 | non | [878219 | +t 
Bel Feber: || 22" |) eens | 188 
59 | “ster | 2-43 || “eraoo | 18 
60 | 9.816943 | ~-4* || 9.877780 
‘ | Cosine. | D. 1". | Sine. Derk": 


Tang. 


9, 923814 
- 924070 
- 924327 
924583 
924840 
925096 
925852 
. 925609 
925865 
926122 
. 926378 

9.926634 
. 926890 
927147 
927403 
927659 
927915 
928171 
928427 
. 928684 
. 928940 

9.929196 
929452 
-929708 
929964 
- 930220 
930475 
. 930781 
. 980987 
. 931243 
. 931499 


9.931755 
932010 
932266 
932522 
932778 
933033 
933289 
983545 
. 933800 
934056 | 

9.934311 
984567 
934822 
935078 
935333 
935589 
935844 
936100 
936355 
936611 


9.936866 
987121 
IBI377 
. 937632 
937887 
. 988142 
. 938398 
. 938653 
. 938908 

9.939163 


Cotang. 


Del”. 


SWWMWMNWNWNWNWWW WW 


Wrewrwaraorw wa 


MEO ADMIT AE AEA a A AAI OA AA 


CX) 


We 


Wd 2 
Or » 3 


I 29 29 29 29 29 2 2929 
SKRLSRRLRSB 


| 
| 


9 
_ 
2 


| Cotang. 


10.076186 
075930 
075673 
075417 
075160 
074904 
074648 
074391 
074135 
073878 
073622 

10.073366 
073110 
072853 
072597 
072341 
072085 
071829 
071573 
071316 
071060 

10.070804 
070548 
070292 
070036 
069780 
069525 
069269 
069013 
068757 
068501 

10.068245 
067990 
067734 
067478 
067222 
066967 
066711 
066455 
066200 
065944 


10.065689 
.065433 
.065178 
-064922 
. 064667 
064411 
.064156 
.063900 
.063645 
063389 

10.063134 
062879 
062623 
062368 
.062113 
.061858 
.061602 
.0613847 
.061092 

10.060837 


Tang. 


El | 


—_ 


fol pes et 
wre 


SOMDIAMTIP OD 


Sine. 


9.816943 
.817088 
817233 
817379 
817524 
.817668 
.817813 
.817958 
.818103 
818247 
.818392 
9.818536 
.818681 
818825 
818969 
819113 
819257 
.819401 
.819545 
.819689 
819832 
9.819976 
820120 
820263 
820406 
820550 
820693 
820836 
820979 
821122 
821265 


9.821407 
.821550 
.821693 
.821835 
.82197'7 
.822120 
.822262 
.822404 
,822546 
. 822688 

9.822830 
822972 
823114 
823255 
. 823397 
. 823539 
.823680 
823821 
.823963 
.824104 


| 9.824245 
824386 
824527 
824668 
824808 
824949 
825090 
825230 
825371 
| 9.825511 


Cosine. 


D..1". 


| 
| 


ITWWNWNWWW 


SESS See eee SERee ESE EEE 
NAHDHHDSDHS SHSSSSSSOW SHNSOHWHNSHHNO 


WNWNWWWWWNWNVMY WNWNVNNNWVNYW WHNVYW 


CQO QO WW BWW 


WWNWNWNWNWWWWM WWNWNWNWNWWNWWW 
WOWWWWWWWD WWWWWWWWtt WWwWWWwWWWKWRWWwWWw 
G2 OV 09 CLOT OS OTOUCOT OT OTS OLOUTSE NI OTSA AIRINNWNID VRNDD 


wrwwrwwwrwre 


Cosine. 


9.877780 
877670 
877560 
877450 
877340 
877/230 
877120 
877010 
876899 
876789 
876678 

9.876568 


id ded 


-Ol006 


9.875459 
.875348 
875237 
875126 
875014 
874903 


874680 
874568 
874456 
9.874344 
874232 
874121 
874009 
873896 
873784 
87367 
873560 
873448 
873335 
9.873223 
873110 
872998 
872885 
87272 
872659 
872547 
1872434 
872321 
872208 
9.872095 
871981 
.871868 
| ..871755 
| .871641 
| .871528 
871414 
871301 
| .871187 
9.871073 


Sine. 


874791 | 


Didl'’s 


ia a a a Drm mck mk fr fh feed fed fend fred fh mech fel forme fom feemh frrmh femed fered feed fod Pr pre pre pr fee femal femme frm fk fom fm fmt fom fmt fc fc fom fumed fom femme mh fom frm fem emma form fol frome fork feel from 
3 


TABLE XXV.—LOGARITHMIC SINES, 


Tang. 


9.939163 
. 939418 
. 939673 
. 939928 
. 940183 
. 940439 
940694 
. 940949 
941204 
941459 
941713 


9.941968 
942223 
942478 
-942733 
942988 
- 943243 
- 943498 
943752 
944007 
944262 


-944517 
944771 
- 945026 
- 945281 
- 945535 
945790 
-946045 
946299 
946554 
946808 


co 


| 9.947063 


.947318 
947572 
947827 
. 948081 
. 948335 
. 948590 
. .948844 
. 949099 
949353 


9.949608 


. 949862 
. 950116 
. 950371 
- 950625 
- 950879 
.951133 
.951388 
. 951642 
. 951896 
9.952150 
- 952405 
952659 
952913 
. 953167 
. 953421 
953675 
- 953929 
. 954183 
9.954437 


Cotang. 


529 29 


OL OLCOTT OURS OT OT OU OT 


WWW WWW 


4.25 


WMNWNW WWNONwwnwnw 
CO OLd9 Ot OOS O19 OF OL 09 


Wrwwmwwwwwwyw w ds 
WOW ow we sreaee Ot Sie 


2 9 29 


SSSR 


WWWWW 
Co G8 Oo Go OO 


Cotang. 


10.060837 
.060582 
0603827 
-060072 


.059817 | 


059561 
059306 
059051 


058796 
058541 


058287 


10.058032 


057777 


VOC 
-057522 


057267 | 


.057012 
056757 
. 056502 
.056248 
.055993 
055738 
10 .055483 
-055229 
054974 
054719 


-054465 | 
-054210 - 


058955 


053701 | 
053446 | 
053192 | 


10.052937 
052682 
052428 
-052173 
.051919 
-051665 
.051410 
.051156 
050901 
-050647 

10.050392 
.050138 
049884 


.049629 | 


049375 
-049121 
048867 
.048612 
048358 
048104 


10.047850 


047595 
-047341 
047087 


046833 
046579 
046325 
046071 

045817 


10. 045563 


Tang, 


COMA WWH OS 


—" 


or 


rN 
CO 


Sine. 


9.825511 
.825651 
825791 
825981 
826071 
-826211 
.826351 
826491 
826631 
826770 
.826910 

9.827049 
-827189 
827328 
827467 
827606 
827745 
827884 
828023 
.828162 
828301 


9.828439 
82857; 
828716 
828855 
828993 
829131 
829269 
829407 
829545 
1829683 

29821 

829959 

830097 

880234 

.830372 

. 830509 

830646 

830784 

.830921 

831058 


| 9.831195 


831332 
831469 
.831606 
831742 
831879 
832015 
832152 
832288 
832425 


| 9°832561 


832697 
832833 


832969 


833105 | 


833241 
833377 
833512 


983648 
9.833783 


Cosine. 


| 
| 


wd 
icy) 
Oo 


WWWWW WWNWWOWNWWNWWW 
LOW wCWwWwWWwWwWWwWoN WC 
Ooo SCOoCocooowonw 


2.30 


2 0 Ww 
22 
@ 


BODO 20 20 
% 


De i | Cosine. 


9.871073 


| .870960 


870846 
870732 
870618 
870504 
870390 
870276 
«870161 
870047 
.869933 


9.869818 
869704 
869589 
869474 
869360 
869245 
.869130 
.869015 
.868900 
868785 


9.868670 
. 868555 
.868440 
868324 
868209 
868093 
.§07978 
. 867862 
867747 
8676381 

9.867515 
.867399 
867283 
.867167 
.867051 

|| .866935 

|| .866819 

.866703 

|| .866586 

.866470 


|] 9.866353 
|| 866237 
|| 866120 
|) 866004 
865887 
86577 
865653 
865536 
865419 
865302 
9.865185 
865068 
864950 
864833 
864716 
864598 
864481 
864363 
864245 
9.864127 


Sine. 


| 


DL: 


Tang. 


=) 
(ou) 


9.954487 
954691 
954946 
. 955200 
955454 
. 955708 
955961 
956215 
- 956469 
956723 
956977 

9.957281 
957485 
957739 
957993 
958247 
. 958500 
958754 
. 959008 
959262 
- 959516 

9.959769 
. 960023 
960277 
960530 
960784 
. 961038 
. 961292 
. 961545 
.961799 
. 962052 

9.962806 
962560 
. 962813 
. 962067 
. 963820 
. 963574 
. 963828 
. 964081 
. 964335 
964588 


. 964842 
. 965095 
965349 
. 965602 
. 965855 
. 966109 
. 966362 
- 966616 
. 966869 
967123 


9.967376 
967629 
.967883 
. 968186 
. 968359 
968643 
. 968896 
. 969149 
969403 

9.969656 


Cotang. I'D: 137 | 


SAA ALAA A AAA AA AAA AHL AAD ALAA A AAA DAE PAAAALRA BARBRA 
WMWWWDW WW NWNW NNN NWYW WNWNWNWNWNVWNWNYW WNHWwNwwwwwn»n wwrwr 
Wwe HBSVSSSSBVES SYRSRERRRY GRGRVVRVRLY CRBS 


oc 


Cotang. 


10.045563 
.045309 
.045054 
.044800 
044546 
044292 
.044039 
043785 
.043531 
043277 
048023 

10.042769 
042515 
042261 
042007 
041753 
.041500 
041246 
.040992 
.0407 -8 
040484 


10.040281 
.039977 
.039723 
039470 
.039216 
038962 
.038708 
038455 
038201 
037948 


10.037694 
.037440 
037187 
-036933 
.036680 
036426 
036172 
035919 
035665 
035412 


10.035158 
.034905 


034651 | 


034398 
034145 
033891 
033638 
033384 
033131 
032877 
10.032624 


0323711 
032117 


031864 


031611 
031357 


.031104 
.030851 
0380597 


10.030344 


Tang. 


CMOIRDUIP WMHS | 


Sine. 


9.833783 
833919 


834054 


834189 
834325 
834460 
834595 
.834730 
834865 
.834999 
835134 
9.835269 
.835403 
835538 
835672 
835807 
835941 
.836075 
836209 
836343 
836477 
9.836611 
836745 
836878 
837012 
837146 
837279 
887412 
837546 
837679 
837812 
9.837945 
838078 
838211 
838344 
8388477 
. 838610 
838742 
838875 
839007 
839140 


9.839272 
839404 
* 839536 
839663 
839800 
839932 
840064 
840196 
840328 
840459 


9.840591 
840722 
840854 
840985 
841116 
841247 
841378 
.841509 
.841640 

9.841771 


Cosine. 


Dai". 


SIOWUVNNWY VYNVNVNVNVNYYNY VVVNVVNVNLYNNY WKVNVNMNYYNVKVVY 
BS OD OND HOOD ODD ICD TCD OTD OUT OO MANION ST 


2% 
SOSCS SCHWOCHWOCKWNWNWNWYHW 


wWwWwN 


~) 


20 


DWWWWWW WWMWNWW WWW 


Cosine. 


9.864127 
.864010 
863892 
863774 
863656 
863538 
.863419 
.863301 
.863183 
863064 
862946 


| 9.862827 


862709 
.862590 


862471 


862353 
862284 
862115 
.861996 
-861877 
861758 


9.861638 
-861519 
.861400 
861280 
.861161 
861041 
860922 
860802 
860682 
860562 


9.860442 
860322 
860202 
860082 
859962 
859842 
859721 
859601 
859480 
859360 

9.859239 
859119 
858998 
858877 
858756 
858635 
858514 
858393 
858272 
858151 

9.858029 
857908 
857786 
857605 


857543 
857422 


857300 
85717 
857056 


| 9.856934 


Sine. 


Dr". 


98 


Seoooeoe 
lane oan ie oes) 


WWWWW WNWWBDEHDWBEHVWHH WEP eH HH Eee 
2 
Sssé S 


eooooo 
YOOOoOooO 


eosscoo 
SD DWNWWD 


CWO WWMOMOIM Wa 


09, 02.09 8 WW WW HW 0 WW 


Seesegeos9o 


o 


TABLE XXV.—LOGARITHMIC SINES, 


Tang. 


9.969656 
. 969909 
. 970162 
970416 
. 970669 
- 970922 
971175 
-971429 
. 971682 
972188 

9.972441 
972695 
972948 


973454 
.I3707 
973960 
974218 
.974466 
974720 
9.974973 
975226 
975479 
975732 
975985 
976238 
.976491 
976744 
.976997 


9.977508 
977756 
978009 
. 978262 
978515 


979021 
979274 
979527 
979780 


. 980286 
. 9805388 
980791 


981297 
.981550 
.981803 
. 982056 
. 982309 


9.982562 
. 982814 
. 983067 
. 983320 
. 988573 
. 983826 
. 984079 
. 984332 
. 984584 
9.984837 


Cotang. 


971985 | 


.973201 


977250 | 


978768 | 


9.980033 


-981044 | 


D. 1”. 


| 
| 


SWWNWNWNNNWNWWW 
SESSSSSLCSIS 


Mow 
0 0 09 


wWwmnnney»y 
GARGS PR) 


IMWMW WOWNHWWOWYM 


ww wo 


4.22 


ri 
g 


2 
] 


ri 
PR 


wWrmdwwwwwnwwye ww 
RRS SSRBSSSRSE RE 


4 


WHWwND 


Cotang. 


| 10.030344 | 
030091 | 
.029838 
029584 | 
.029331 | 


.029078 


028825 


028571 


028318 


028065 


027812 
10.027559 | 


027805 | 
027052 
.026799 


.026546 
026293 
026040 
025787 
025534 


-025280- 
10.025027 


02477 


024521 | 
024268 | 
024015 | 
023762 | 
023509 | 


023256 


.028003 | 


022750 
10.022497 
022244 
.021991 
021738 


.021485 
.021232 | 
020979 


020726 
020478 
020220 
10.019967 


.019714 | 
.019462 
.019209 | 


018956 
018703 
018450 


018197 
017944 
017691 


10.017438 


.017186 


016933 | 
.016680 


.016427 
.016174 
015921 
.015668 

015416 


10.015163 


or 
or 


g 


~ 
lor) 


(SV) 
ies} 


coq te 
He OUS> 


[sy] 
So 


ta) 
© 


nil emul aed 


| 


ie =) 


Sine. 


.841902 
.842033 
.842163 
.842294 
842424 
. 842555. 


.842815 
.842946 
.843076 
9.843206 


COIS) Or 


| 


. 843466 
. 843595 
843725 
. 843855 
.843984 
. 844114 
844243 
844372 
9.844502 
844631 
.844760 
.844889 
.845018 
.845147 
845276 
.845405 
. 845533 
/845662 
9.845790 
.845919 
.846047 
.846175 
. 846304 
846432 
846560 
.846688 
.846816 
846944 
847071 
.847199 
847327 


=) 


847582 


847964 
848091 


© 


848345 
848472 
“848599 
848726 
“848852 


849232 
849359 


| 9.841771 | 


. 842685 | 


.843336 | 


847454 
847709 | 
847336 | 


848218 | 


09 0 WW WNWWWWNHWNKNWW WWWNWWNWWNWWW WWNUWNWNWNMNWMWNWW WNWWNWWWNWWwW®D wi ri 


: 848979 | 
57 | .849106 


® WW 


9.849485 | 


Cosine. 


Cosine. Tang. Cotang. 
9.856934 9.984837 4.29 10.015163 
.856812 . 985090 ¥ 90 .014910 
.856690 . 985343 4 "99 014657 
.856568 . 985596 a 20 014404 
.856446 . 985848 rg ae 014152 
.856323 .986101 ry 99 .013899 
.856201 986354 4 ee .013646 
.856078 . 986607 ry Re .013393 
.855956 .986860 A. S .0138140 
855833 987112 | 4°55 .012888 
.855711 - 987365 4 ; 29 .012635 
9.855588 9.987618 | 4 99 | 10.012382 
855465 987871 4 20 .012129 
855342 . 988123 vy 29 .O11877 
855219 . 988376 4.99 .011624 
855096 . 988629 a. 99 .0113871 
854973 . 988882 ia 20 .011118 
-854850 . 989134 4 "99 .010866 
.854727 . 989387 4 Pie .0106138 
854603 -989640 | 7°55 010360 
854480 .989893 490 010107 
9.854356 9.990145 4.99 10.009855 
854233 . 990398 ra pa .009602 
.854109 . 990651 re 20 .009349 
.853986 -990903 | 4°55 009097 
853862 . 991156 ae as . 008844. 
.853738 . 991409 ae 99 008591 
853614 .991662 4.90 008338 
853490 “91914 | 7°55 .008086 
. 853366 . 992167 4.99 007833 
853242 . 992420 4.20 007580 
9.853118 9.992672 | 4 gq | 10.007828 
.852994 | * 992025 | 45s 007075 
. 852869 3. .993178 ‘As 39 .006822 
852745 3: 993431 4. 50 .006569 
852620 oy . 993683 vig 99 .006317 
.852496 sy . 993936 4 99 . 006064. 
852371 a . 994189 a 20 .005811 
852247 on .994441 4. Bes .005559 
852122 9 : . 994694 ay 29 005306 
.851997 0. . 994947 4 "30 . 005053 
9.851872 | -995199 | 4 99 | 10.004801 
SSOLV afar: 995452 | 459 004548 
.851622 ee . 995705 a an . 004295 
.851497 9 ; . 995957 4 "99 .004043 
.851372 9. . 996210 4. 99 .003790 
.851246 3. . 996463 re 20 .003537 
.851121 o .996715 4 ‘99 . 003285 
.850996 pa . 996968 re Se .0038032 
.850870 . 997221 as 20) 00277 
850745 oe . 997473 a 99 002527 
9.850619 | 5 9.997726 | 4 59 | 10.002274 
. 850493 9 : | 997979 4. 350 002021 
850368 3 . 998231 Ps 96 .001769 
850242 | %: .998484 | 7°55 001516 
.850116 | 5 998787 | 7 °S5 001263 
. 849990 Ss . 998989 4 sese .001011 
849864 | %° -999242"| 7°55 000758 
. 849738 ae . 999495 4 30 .000505 
849611 | 5° 999747 | 4°50 000253 
9.849485 | ; 10 .000000 a 10.000000 
Sine, Cotang. ! D. 1” Tang. 


SCHWWRPUAMNIMDLO 


- 


TABLE XXVI—LOGARITHMIC VERSED SINES. 


0° 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


° 


COMNIAIP WMHS 


OU 


or 


So Or 


Vers. 


\Inf. neg. 
2.626422 
13228482 


.580665 


3.830542 


024362 
182725 
316618 
.432602 
534907 
626422 
709207 
784784 
854308 
918678 
978604 
034661 
087319 
136966 
183928 
228481 
250859 
311266 
349876 
- 386843 
422300 
456367 
489148 
520736 
551216 
580662 


.609143 
.636719 
663447 
.689377 
~714555 
739023 
. 762821 
. 785985 
. 808547 
.830537 
.851985 
.872915 
. 893353 
. 913322 
. 982841 
~9519381 
.970611 
. 988898 
.006807 
. 024355 


041555 | 


058421 
.074965 
.991201 
107138 


5 | .122789 
= 


. 138162 
153268 
168116 


6.182714 


q — 2l | Ex. sec. 
9.070 
120 || 120 Inf. neg. 
120 | 120 2.626422 
| 120 || 120 8.228482 
| 120 || 120 .580665 
| 120 || 120 3.880542 
120 || 120 |4.024363 
120 || 120 | .182725 
120 || 120} .816619 
120 || 121 | .432603 
119 |} 121 | .584908 
119 || 121] .626424 
119 || 122 4.709209 
119 || 122 | .784787 
119 |} 122 | .854812 
119 || 123} .918681 
119 || 123 '4.978608 
119 || 124 |5.0384666 
119 |) 124} .087825 
119 || 125] .136972 
119 || 125} .183935 
119 || 126 | .228488 
118 || 126 5.270868 
118 || 127 | .3811275 
118 || 128} .349886 
118.|/ 129} .886854 
118}| 129 | .422312 
118 || 180} .456379 
118 || 181} .489161 
117 || 182 | .520750 
117 || 188} .5512381 
117 || 184] .580679 
117 || 134 5.609160 
117 || 185 | .636738 
116 || 186} .663467 
116 || 187 | .689398 
116 || 188) .714577 
116 || 140} .7389047 
116 ||} 141 | .762847 
115 || 142 | .786012 
115 || 143 | 80857 
115 || 144 | .830567 
115 || 145 |5.852016 
114 || 147 | .872948 
114 || 148) .893387 
114 || 149] 913357 | 
114 || 151} .98287 
113 || 152 | .951970 
113 || 154] 970652. 
113 || 155 |5 988940 
112 || 157 6.006851 
112 || 158) .024401 || 
| 112 || 160 6.041602 
} 111 || 161} .058470 
/111]/ 163} .075017 
111 || 164} .091254 
110 || 166| .107194 
1 110]/ 168) .122846 
110 || 169 | .1388222 
109 || 171 | .153880 
109 || 173 | .168180 
109 |) 175 |}6.182780 


| 6480 


| 6840 
|, 6900 


4} 


3600 
3660 
3720 
3780 
3840 
8900 
3960 
4020 
4080 
4140 
4200 
4260 
4320 
4380 
4440 
4500 
4560 
4620 
4680 
4740 

4800 
4860 
4920 
4980 
5040 
5100 
5160 
5220 
5280 
5340 
5400 


5460 
5520 
5580 
5640 
5700 
5760 
5820 
5880 
5940 
6000 | 


6060 
6120 
6150 
6240 
6300 
6360 
6420 


6540 
6600 
6660 
6720 
6780 


6960 
7020 
7080 
7140 


_ 
ar CSCOMDIMHDOUPWWH © 
ao 


a 


SUR) 


(or) for) 


a 


7200 
0 


Vers. 


.182714 
.197071 
.211194 
.225091 
28877 

. 252236 
. 265497 
.278558 
-291426 
.3804106 
.3816603 
. 828923 
841071 
. 803052 
. 864869 
.316528 
. 3888032 
.399386 
.410593 
.421657 
.482583 
6.448372 
.454029 
.464557 
.474959 
-485238 
.495396 
.505438 
.515364 
.525178 
- 5384882 
.544480 
.553972 
. 563362 
.572651 
.581842 
. 590936 
.599937 

. 608845 
.6176638 
. 626392 
. 6850384 
.643591 

. 65264 | 
. 660456 
. 668767 
. 677000 
.685155 
. 693234 
. 101239 
709171 
\6.717030 
. (24820 
. 732540 
.740192 
CATT 
155297 
. 762752 
.770144 
107403 


6.784741 


| 175 | 


lor) 


EX. sec. 


6.182780 
197139 
211264 
225164 
288845 
252314 
265577 
278641 
291511 
304193 
316693 

6.329016 
341167 
853150 
364970 
376631 
388138 
399494 

410705 
421772 
432700 

6.443498 
454158 
464684 
475089 
48537 
495532 
505577 
515506 
525324 
535031 


6.544632 
.554128 
563521 
572813 
.582008 
.591106 
.600110 
.609021 
.617843 
626575 

6.635221 
643782 
652259 
. 660655 
. 668970 
677206 
685365 
. 693448 
. 701457 
709393 


T7257 


725050 
7382775 
740431 
. 748020 
. 755544 
. 763004 
. 770400 
777733 


6.785005 


AND EXTERNAL SECANTS. 


ny 


i SS 


Wan 


} 


7260 
7320 


9 
~ 


7380} 3} 


7440| 4 


7500) 5} . 
7560| 6) . 


7620| 7 


7680| 8| . 


7740| 9 
7800 10 
7860) 11 


792012) . 


7980 13) °. 


8040 | 14 
8100 | 15 
8160 | 16 
8220| 17 
8280/18 


10080 | 48 
10140) 49 
10200) 50 


10260) 51 | 


10820 | 52 


10380) 53] 


10440) 54 
10500. 55 
10560 56 


10620 57} 


10680: 58 
10740 59 
10800 

u 


SEE 


7200) 0 6.784741 | 
hy 


60) 
é 


| | Ex. sec. 


16.'785005 
792217 
.799370 
. 806464 
813501 
.820482 
. 827406 
83427 
.841093 
847857 
854568 
| .861228 
.867837 
874396 
.880907 
.887369 
.893783 
.900151 
. 906472 
.912748 
.918979 


925165 
. 931308 
. 937408 
943465 
.949480 
955455 
.961388 
967281 
9731385 
978949 
984725 
990463 
996164 
001827 
007454 
013044 
.018599 
024119 
029604 
035054 


040471 
045854 
051204 
056522 
.061807 
067061 
.072282 
077473 
082633 
087763 
3 | 7.092862 
3} .097932 
102978 
107985 
112968 
117922 
. 122849 


1277 


132619 


Abas 
7.137464 


WOOsIOR WMH © 


ba 3 


~ 


Vers. 


7.136868 
.141679 
. 146464 
151222 
155954 
. 160661 
. 165342 
. 169998 
174630 
. 179286 
. 183819 
188377 
192912 
197423 
.201910 
-206375 
210817 
215286 


219683 


224007 

228360 
7.232691 
237000 
241288 
"245555 
249801 
254027 
258232 
262416 
266581 
210726 
274851 
278056 
283043 
287110 
291158 
295187 
299197 
303190 
307164 
311119 
315057 
318977 
322880 
326705 
330632 
334483 
338316 
342133 
345933 
349716 
853483 
357233 
360968 
364686 
368389 
372076 


30 |7.386668 


~t 


=I 


S84 


=) 


= 


Ex. sec. 


7.137464 
142281 
147072 
151837 
156577 
. 161290 
.165978 
170641 
175279 
179893 
. 184483 


7.189048 
193589 
198108 
202602 
200074 
211523 
215949 
220853 
224785 
229095 


238433 
237750 
242046 
. 246320 
250574 
254807 
259019 
- 263212 
267384 
201587 


275669 
279783 
283877 
287952 
292007 
296045 
800063 
804068 
808045 
.312009 


815955 
.319883 
.823794 
327687 
.3831563 
.890422 
839263 
843089 
846897 
850689 


"354464 
| , 808228 | 
.861966 

9 | .365693 
>| .869404 
.3873100 
.3876780 
880444 
. 884094 
’ 887728 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


4° 5° 
Vers. | D. 1",.| Ex. see. | D. 1" / Vers. D. 1°. | Ex. see.|D, 1” 
0 | 7.386668 | 60.17 | 7.387728 | 60.32 0 | 7.580389 | 48.15 |7.582045 | 48.33 
1 .890278 | 59.938 .3891847 | 60.07 1 .583278 | 47.98 | .584945 | 48.17 
2 .398874 | 59.67 .394951 | 59.82 2 .586157 | 47.82 | .587835 | 48.00 
3 .897454 | 59.48 .898540 | 59.57 3 -589026 | 47.67 | .590715 | 47.87 
4 -401020 ; 59.18 402114 | 59.33 4 .591886 7.52 | .598587 oul | 
5 .404571 | 58.93 .405674 | 59.10 5 594737 | 47.35 | .596449 | 47.53 | 
6 .408107 | 58.70 .409220 | 58.85 6 .597578 | 47.20 | .599301 | 47.38 
ve -411629 | 58.47 .412751 | 58.62 v6 . 600410 7.05 | .602144 7.25 
8 .415187 | 58.23 .416268 | 58.38 8 . 603233 | 46.90 | . 604979 7.08 
9 .418631 | 58.00 .419771 | 58.15 9 -606047 | 47.7: .607804 | 46.92 
L077 42208Y 15077 .423260 | 57.92 || 10 .608851 | 46.60 | .610619 | 46.78 
11 | 7.425577 | 57.53 | 7.426735 | 57.70 || 11 | 7.611647 | 46.43 |7.613426 | 46.63 
12 .429029 | 57:30 .430197 | 57.45 || 12 .614433 | 46.30 | .616224 | 46.48 
13 .4382467 | 57.08 .433644 | 57.25 || 18 -617211 | 46.15 | .619013 | 46.35 
14 .435892 | 56.85 .437079 | 57.00 || 14 .619980 | 45.98 | .621794 | 46.18 
15 .439303 | 56.63 .440499 | 56.80 || 15 -622739 | 45.87 | .624565 | 46.05 
16 .442701 | 56.42 .443907 | 56.57 16 -625491 | 45.7 .627328 | 45.90 
17 .446086 | 56:20 -447301 | 56.35 17 .628233 | 45.57 | .630082 | 45.75 
18 .449458 | 55.97 .450682 | 56.13 18 .630967 | 45.42 | .632827 | 45.62 
19 -452816 | 55.77 .454050 | 55.92 19 .633692 | 45.28 | .635564 | 45.48 
20 .456162 | 55.55 .457405 | 55.7% 20 .636409 | 45.13 | .638293 | 45.33 
21 | 7.459495 | 55.33 | 7.460748 | 55.48 || 21 | 7.639117 | 44.98 |7.641013 45.18 
22 .462815 | 55.42 .464077 | 55.28 || 22 .641816 | 44.87 | .648724 | 45.07 
23 .466122 | 54.92 .467394 | 55.08 || 23 .644508 | 44.72 | .646428 | 44.90 
24 .469417 | 54.70 -470699 | 54.87 || 24 -647191 | 44.57 | .649122 | 44.78 
25 .472699 | 54.50 .473991 | 54.65 || 25 -649865 | 44.45 | .651809 | 44.65 
26 .475969 | 54.28 477270 | 54.47 |) 26 -652532 | 44.30 | .654488 | 44.50 
27 .479226 | 54.10 .480538 | 54.25 || 27 .655190 | 44.17 | .657158 | 44.37 
28 .482472 | 53.88 .4838793 | 54.05 28 .657840 | 44.05 | .659820 | 44.93 
29 -485705 | 53.7 .487036 | 53.85 |} 29 .660483 | 43.90 | .662474 | 44.12 
30 .488927 | 53.48 -490267 | 58.67 || 30 .663117 | 48.77 | .665121 | 43.97 
31 | 7.492136 | 53.28 | 7.493487 | 53.45 || 81 | 7.665743 | 43.63 |'7.667759 43 .83 
82 495333 | 53.10 .496694 | 53.27 || 32 .668361 | 43.50 | .670889 | 43.7: 
33 498519 | 52.90 .499890 | 53.07 || 33 .670971 | 43.38 | .673012 | 43.57 
34 501693 |} 52.7% .503074 | 52.88 || 34 .678574 | 43.23 | .675626 | 43.45 
35 504856 | 52.52 006247 | 52.68 385 .676168 | 43.12 | .678233 | 43.33 
36 508007 | 52.33 .509408 | 52.50 || 36 .678755 | 42.98 | .680833 | 43.18 
37 511147 | 52.13 .512558 | 52.32 || 37 .681834 | 42.87 | .683424 | 43.07 
38 -614275 | 51.95 .015697 | 52.12 || 38 .683906 | 42.73 | .686008 | 42.95 
39 517392 | 51.7 .618824 | 51.93 389 .686470 | 42.60 | .688585 | 42.89 
40 -620498 | 51.58 -621940 | 51:77 40 .689026 | 42.48 | .691154 | 42.68 
41 | 7.523593 | 51.40 | 7.525046 | 51.57 || 41 | 7.691575 | 42.35 7.698715 | 42.57 
42 526677 | 51.22 .528140 | 51.38 || 42 .694116 | 42.%3 | .696269 | 42.43 
43 529750 | 51.03 081220 |°51.22°'| 43 .696650 | 42.12 | .698815 | 49.33 
44 532812 | 50.85 .534296 | 51.02 44 .699177 | 41.98 | .701855 | 42.90 
45 .5385863 | 50.68 .537857 | 50.85 || 45 |. .701696 | 41.87 | .'703887 42.07 
46 .538904 | 50.50 .540408 | 50.68 || 46 - 704208 | 41.73 | .706411 | 41.97 
ik .541934 | 50.32 .543449 | 50.50°|| 47 | .706712 | 41.63 | . 708929 | 41.83 
48 .5449538 | 50:15 .546479 | 50.33 48 - 709210 | 41.50 | .711439 | 44.72 
49 .547962 | 49.98 .549499 | 50.15 || 49 .711700 | 41.88 | .718942 | 41.60 
50 .550961 | 49.80 .552508 | 49.98 || 50 714183 | 41.27 | .716438 | 41.48 
51 | %.553949 | 49.63 | 7.555507 | 49.80 || 51 | 7.716659 | 41.15 |'7. 718997 41.37 
52 .556927 | 49.48 .558495 | 49.65 52 .719128 | 41.03 | .721409 | 41.95 
53 .559895 | 49.28 .561474 | 49.47 || 53 .721590 | 40.92 | .723884 | 41.13 
54 .562852 | 49.13 .564442 | 49.32 || 54 . 724045 | 40.80 | .726352 | 41.02 
55 -565800 | 48.95 .567401 | 49.13 55 -726493 | 40.68 | .728813 | 40.90 
56 .568737 | 48.80 .5703849 | 48.98 56 728934 | 40.57 | .731267 | 40.7 
57 .571665 | 48.63 .5738288 | 48.82 || 57 . 731368 | 40.47 | .733714 | 40.68 
58 5745838 | 48.47 .576217 | 48.65 || 58 . 733796 | 40.388 | .786155 | 40.57 
59 .577491 | 48.30 .579136 | 48.48 59 -736216 | 40.23 | .738589 | 40.45 
7.580389 | 48.15 | 7.582045 | 48.33 || 60 | 7.738630 | 40.13 |'7.741016 40.33 


| 


DAS mMeWMS | ~ 


Pee a I en i ee a ae emcee oa ERE a al SE ca EN RAMI ERRATA DRA a 
CWJCW ICH OCH an Cru a ce g 
CPG 02 = t oe tS Se ”) < 


Vers. 


7.738630 
. 741038 
. 748488 
. 745832 
. 748219 
750600 
- 752974 
(55842 
757703 
-'760058 
762406 
7.764749 
767084 
. 769414 
ART 
774054 
776365 
- 778670 
- 780968 
783261 
785547 
. 787828 
. 790102 
792371 
.794633 
.796890 
.799141 
.801885 
. 803625 
.805858 
808086 


.810308 
.812524 
.814734 
.816939 
.819139 
.8213382 
. 823521 
.8257038 
.8278380 
. 8380052 


| 7.932218 


. 


“J 


~I 


“851475 


7.853589 
.855697 
.857800 
.859898 
.861991 
864079 
866162 
. 868240 
.870313 

% 872381 


Ex. sec. 


| 40.13 | 7.741016 


748486 
. 745850 
. 748258 
~750658 
"753052 


. 755440 


(57821 
760196 
762565 
764927 
T67282 
- 769632 
T1975 
774312 
. 776643 
778968 
. 781286 
. 783599 
785905 
~ 788206 
7.790500 
792789 
795071 
797348 
799619 
801884 
804143 
806397 
808644 
810886 


7.813123 
.815353 
.817578 
.819798 
. 822012 
. 824220 
826423 
. 828620 
. 830812 
.882999 

7.885180 
. 837356 
.839526 
.841691 
.843851 
. 846005 
.848155 
.850299 
.852437 
.854571 


856700 
858823 
.860942 
.863055 
865163 
867266 
869365 
871458 
873546 
7.875630 


=z 


~ 


AND EXTERNAL SEUANTS. 


Vers. 


| 7.872881 


874444 
876502 
878555 
880603 
882647 
. 884686 
886720 
888749 
89077 
892793 


7.894808 
896818 
898824 
- 900825 
902821 
.904813 
- 906800 
908783 


-910761 | 


912735 


7.914704 
. 916668 
. 918629 
. 920584 
922536 
924483 
926425 
. 928364 
930297 
982227 


7.984152 
936073 
. 937990 
. 939903 
941811 
948715 
. 945615 
947511 
-949402 
951290 


7.953173 
955052 
.956928 
958799 
. 960666 
- 962529 
. 964388 
966243 
. 968094 
969941 


7.971785 
973624 
975459 
977291 
.979118 
.980942 
982762 
984578 
. 986391 

7.988199 


Ex, sec. | 


% 875630, | 
877708 
879782 
881851 
883915 
885974 
888029 
890078 
892123 
894164 
896199 


7.898280 
. 900256 
902278 
904295 
. 906307 
-908315 
.910319 
912317 
.914312 
. 916802 


7.918287 
920268 
922245 
924217 
926184 
. 928148 
.930107 
. 932062 
. 934012 
.935958 


7.937900 
939838 
941772 
.948701 
. 945626 
947547 
949464 
.951376 
958285 
955189 


7.957090 
. 958986 
. 960878 
962767 
. 964651 
. 966531 
- 968408 
- 970280 
972148 
.974018 


7. 975874 
977730 
979583 
.981432 
9838277 
. 985119 
. 986956 
988790 
. 990620 


30.08 17.992446 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


<—lS 
8° 9° 
Vers. | D.1’. | Ex. sec. | D. 1’. ||’ Vers. | D.-1". | Ex. see./D. 1°, 
7.988199 | 30.08 | 7.992446 | 80.388 || 0 | 8.090317 | 26.72 |8.095697 | 27.05 
.990004 | 380.02 - 994269 | 30.82 1 -091920 | 26.68 | .097320 | 27.02 
.991805 | 29.95 .996088 | 30.25 2 .093521 | 26.63 | .098941 | 26.97 
. 993602 | 29.88 .997903 | 30.18 3 .095119 | 26.58 | .100559 | 26.92 
995395 | 29.83 | 7.999714 | 30.13 4 .096714 | 26.52 | .102174 | 26.87 
997185 |.29.7°7 | 8.001522 | 30.07 5 .098305 | 26.48 | .103786 | 26.82 
7.998971 | 29,72 -003326 | 30.00 6 .099894 | 26.43 | .105395 | 26.77 
8.000754 | 29.63 .005126 | 29.95 vg -101480 | 26.40 | .107001 | 26.7% 
.002582 | 29.60 -006923 | 29.88 8 . 103064 | 26.33 | .108605 | 26 67 
-004308 | 29.52 .008716 | 29.83 9 -104644 | 26.28 | .110205 | 26.63 
.006079 | 29.47 .010506 | 29.7% 10 106221 -| 26.25 | .111803 |. 26.58 
8.007847 | 29.40 | 8.012292 | 29.% j 11 | 8.107796 | 26.18 |8.113398 | 26.53 
.009611 | 29.35 .014074 | 29.65 |} 12 .109867 | 26.15 | .114990 |. 26.48 
.011872 | 29.28 .015853 | 29.58 || 13 - 110936 -|| 26.10 | .116579 |. 26.45 
.013129 | 29.22 .017628 | 29.53 || 14 112502 | 26.05 | .118166'|. 26.38 
.014882 | 29.17 .019400 | 29.47 || 15 -114065 |.26.00 | .119749 | 26.35 
.016632 | 29.10 .021168 | 29.42 || 16 115625 |-25.95 | .121330 | 26.30 
.018878 | 29.05 022933 | 29.85 || 17 117182 || 25.92 | .122908 | 26.25 
.020121 | 29.00 024694 | 29.30 || 18 .118787 | 25.87 | .124483 | 26.2% 
.021861 | 28.93 -026452 | 29.23 || 19 120289 | 25.82 | .126056 | 26.17 
028597 | 28.87 | 028206 | 29.18 || 20 | .121888 | 25.77 | .127626 | 26.12 
8025329 | 28.82 | 8.029957 | 29.13 || 21 | 8.128384 | 25.72 |8.129193 | 26.07 
-027058 | 28.75 .081705 | 29.07. || 22 124927 | 25.68 | .130757 | 26.02 
.028783 | 28.70 033449 | 29.00 || 23 126468 | 25.65 | .1382318 | 25.98 
.030505 | 28.65 .035189 | 28.97 || 24 128006 | 25.58 | .188877 | 25.93 
-032224 | 28.58 036927 | 28.90 || 25 129541 | 25.55 | .185433 |. 25.90 
-033939 | 28.53 .038661 | 28.88 || 26 .181074 | 25.50 | .186987 |. 25.85 
.035651 | 28.47 .040391 | 28.78 7 . 182604 | 25.45 | 2188588 |.25.80 
.037359 | 28.42 -042118 | 28.7 28 .184131 | 25.40 | .140086 |. 25.75 
-039064 | 28.37 .043842 | 28.68 || 29 - 185655: | 25.37 | .141681 |. 25.7% 
-040766 | 28.30 | .045563 | 28.62 || 830 | 137177 | 25.82 | .148174 |.95 67 
8.042464 | 28.25 | 8.047280 | 28.57 || 31 | 8.138696 | 25.97 |8. 144714 | 25.63 
.044159 | 28.20 048994 | 28.50 || 32 - 140212 | 25.23 | .146252 | 25.58 
.045851 | 28.13 .050704 | 28.47 || 33 141726 | 25.18 | .147787 | 25.53 
.047539 | 28.08 052412 | 28.40 || 34 148237 | 25.13 | .1493819 | 25.50 
.049224 | 28.03 .054116 | 28.35 || 385 -144745 | 25.10 | .150849 | 25.45 
-050906 | 27.98 -055817 | 28.28 || 36 . 146251 | 25.05 | .152876 | 25.40 
052585 | 27.92 .057514 | 28.25 || 37 147754 | 25.02 | .158900 | 25.37 
054260 | 27.87 -059209 | 28.18 || 38 . 149255 | 24.95 | .155422 | 25.33 
055932 | 27.82 .060900 | 28.13 || 39 -150752 | 24.93 | .156942 | 25.27 
057601 | 27.7 .062588 | 28.08 || 40 152248 | 24.88 | .158458 | 25.25 
8.059266 | 27.72 | 8.064273 | 28.03 || 41 | 8.153741 | 24.88 |8.159973 | 95.18 
-060929 | 27.65 065955 7.97 || 42 .155281 | 24.7 .161484 | 25.17 
062588 7.60 067638 7.93 || 43 .156718 | 24.75 | .162994 | 25.10 
.064244 | 27.55 . 069809 7.87 || 44 . 158203 4.72 | .164500 | 25.07 
.065897 | 27.48 .070981 | 27.82 || 45 .159686 | 24.67 | .166004 | 25.08 
-067546 | 27.45 072650 7.77 |) 46 .161166 | 24.62 | .167506 | 24.98 
-069193 | 27.38 .074316 | 27.4% 47 -162643 | 24.58 | .169005 | 24.95 
-070836 | 27.33 075979 | 27.67 || 48 .164118 | 24.53 | .170502 | 24.90 
072476 | 27.30 077639 | 27.60 || 49 165590 | 24.50 | .171996 | 24.87 
-OV4114 | 27.23 | .079295 | 27.57 || 50 | 1167060 | 24.45 | 173488 | 24.82 
8.075748 | 27.18 | 8.080949 | 27.52 51 | 8.168527 | 24.42 (8.174977 | 24.78 
-OT7379 | 27.18 -082600 | 27.45 || 52 .169992 | 24.37 | .176464 | 24.7% 
.079007 | 27.07 -084247 | 27.42 3 171454 | 24.83 | .177948 | 24.7 
-080631 | 27.03 085892 | 27.37 || 54 172914 | 24.80 | .179430 | 24.65 
082253 | 26.98 087584 | 27.30 || 55 174872 | 24.25 | .180909 | 24.62 
083872 | 26.93 089172 | 27.27 || 56 175827 | 24.20] .182386 | .24.58 
.085488 | 26.87 .090808 | 27.20 v 177279 | 24.17 | .183861 | 24.53 
-087100 | 26.83 -092440 | 27.17 || 58 178729 | 24.13 | .185883 | 24.50 
.088710 | 26.78 -094070 | 27.12 || 59 -180177 | 24.08 | .186803 | 24.47 
8.090317 | 26.72 | 8.095697 | 27.05 || 60 8.181622 | 24.05 18.188271 | 24.42 


~ 


= 
So MIM URwWDWKo| 


ao 


Z 


Vers. 


. | Ex. sec. | 


| 8.181622 | 2 


.183065 
. 184505 
.185943 
.187379 
.188812 


190243 | 23.8 
191671 
-193097 | 2 
.194521 | 2 
195942 


197361 
“DOe78 


"200192 | 2 
“201604 


. 203014 
.204421 
.205826 
. 207229 
- 208630 
.210028 
-211424 
.212318 
.214209 
-215599 
.216936 
.218371 
.219753 
.221183 
222512 
223888 

225261 

226633 
. 228002 
. 229369 


-230735. | 


232097 
233458 
.284317 
236173 


-20102 27 j 


238830 
. 240230 


241578 
242924 


244267 


. 245609 


. 246948 


218286 
219621 | 
250955 

§ 252286 


. 253615 
. 204942 
2562 68 
201) 591 
. 258912 
260231 
. 261548 

- 262863 


60 | $.261176 


) Ww d 


8.188271. | 


1897 36 
. 191198 
-192659 
194117 
195572 
197025 
198476 
. 199925 
-201371 
202815 


8.204257 
205696 
.207133 
208568 
.210001 


211451 


6212859 
.214285 
215708 
217130 
8.218549 
-219966 
221i BRO 
222793 


word 

.224203 
.225611 
. 227017 


222424 
. 229822 
. 231221 
8.232619 
.234014 
. 235407 
.236797 
. 238186 
. 239572 
anand 
2423: 
3449 


.245097 


8.246473 
. 247847 
~ 249219 
. 250589 


. 251957 


. 258822 
. 254686 
.206047 
.257407 
.208764 
8.260120 
.261473 
. 262825 
.264174 
. 265522 
. 266867 
.268211 
.269552 
.270892 
8.272229 


AND EXTERNAL SECANTS. 


eanerzaer > 


(o 2) 


Oacan Meshes 
RA RRB LHKS 


Ex, Sec. | 


'8 .272229 
. 273565 
| .274898 
276230 | 
.277560 

. 278888 
. 280213 
.2815387 
. 282859 | 
. 284179 
. 285498 


'8.286814 
288128 
289441 
290751 

292060 
293367 
294672 
295905 


297276 


"29857 


j0.299873 
.801169 
802463 
3803755 
3805045 
806334 
.807620 
.3808905 
.310188 
.311469 


812749 
.814026 
.815802 
.316576 
.317849 
.819119 
820388 
.821655 
.322920 
.824183 


325445 
329705 
327964 
820220 
830475 
881728 
332980 
334229 
835477 
836724 
'8.337968 
339211 
840453 
841692 
842030 
344166 
343401 
846634 
347865 | 20. 
2 8.349095 | 20.47 


ore 2>HAS 33 ( ( 
z 5 = - G 
OO 


12° 


y Vers. | D. 1’ 
0 | 8.339499 | 20.02 
1 .840700 | 20.00 
2} 3841900 | 19.95 
3 .343097 | 19.95 
4 .844294 | 19.90 
5 .3845488 | 19.88 
6 .846681 | 19.85 
& .847872 | 19.82 
8 .849061.| 19.80 
9 .850249 | 19.7 
10 .301485 | 19.7 
| 11 | 8.352620 | 19.72 
12 .853803..; 19.68 
! 13 004984 | 19.67 
14 .806164 | 19.63 
15 .857342 | 19.60 
16 -3808518 | 19.58 
7 .809693 | 19.55 
18 .360866.} 19.53 
19 .3862083 | 19.50 
20 .868208 | 19.48 
21 | 8.364377 | 19.438 
22 .3805548 | 19.48 
2é .866709 | 19.38 
2! 067872 | 19.37 
25 .869034 | 19.35 
26 .870195 | 19.382 
27 .38113854 | 19.28 
28 872511 | 19.27 
29 310067. | 19.25 
380 .874822 | 19.20 
31 | 8.875974 | 19.18 
32 804125 | 19.17 
28 o1e270 | 19.13 
3¢ .379423.| 19.12 
35 .880570 | 19.08 
36 .881715 | 19.05 
37 .882858 | 19.03 
38 .884000 | 19.02 
39 .885141 | 18.98 
40 .886280 | 18.95 
41-| 8.387417 | 18.93 
42 .888553.| 18.92 
43 .889688 | 18.88 
44 -890821 | 18.85 
45 .891952. | 18.838 
46 .893082 | 18.82 
47 .894211.| 18.4 
48 .895888 | 18.75 
4s -096463 .| 18.7% 
50 397587 | 18.72 
51 | 8.898710 | 18.68 
52 .38998381 | 18.67 
53 .400951 | 18.63 
54 .402069 | 18.62 
55 .408186 | 18.58 
56 -404301 | 18.57 
57 .405415 | 18.53 
58 .406527 | 18.52 
59 .407688 | 18.50 
60 | 8.408748 | 18.47 


Ex. sec. | SOPs ih a 


| 


TABLE XXVI.—LOGARITHMIC VERSID SINES 


13° 


f Vers. | D. 1". | Ex. sec.jD. 1”. 
| | | | 1 

8.849095 | 20.47 || 0 | 8.408748 | 18.47 ‘8.420024 | 18.95 
.350323 | 20.43 || 1] .409856 | 18.43 | .421161 | 18.93 

851549 | 20.42 || 2] .410962 | 18.42 | .422297.) 18.90 

852774 | 20.38 || 3] 412067 | 18.40 | .423431 | 18.88 

.853997 | 20.35 |} 4 | .413171 | 18.88 | .424564.| 18.87 

.855218 | 20.33 || 5 | .414274 | 18.35 | .425696 | 18.83 

856438 | 20.30 || 6 | .415375 | 18.32.| .426926 | 18.82 

857656 | 20.28 || 71 .416474 | 18.30.) .427955.| 18.80 

858873 | 20.25 || 8 | .417572 | 18.28-| .429083.! 18.77 

860088 | 20.22 || 9 | .418669 | 18.25 | .430209 | 18.75 

.861301 | 20.20 |! 10 | .419764 | 18.23 | .431334 | 18.73 

8.862513 | 20.18 || 11 | 8.420858 | 18.22 |8:432458 | 18.70 
863724 | 20.13 || 12 | 1421951 | 18.18 | .423580.| 18.67 

264982 | 20.12 || 13 | 1423042 | 18.17 | .484700.| 18.67 

.366139 | 20.10 || 14 | .424132 | 18.13 | .485820 | 18.63 

867345 | 20.07 || 15 | 1425220 | 18.12 | .436938.| 18.62 

.868549 | 20.03 || 16 | .426307 | 18.10°; .438055 | 18.58 

869751 | 20.02 || 17 | 1427393 | 18.07 | .439170 | 18.57 

870952 | 19.98 ||.18 | 1428477 | 18.05 | .440284 | 18.55 

872151 | 19.95 || 19 | .429360 | 18.02 | .441397 | 18.53 

873348 | 19.95 || 20 | .430641 | 18.02 | .442509 | 18.50 

8.374545 | 19.90 |! 21 | 8.431722 | 17.97 18.443619 | 18.47 
875739 | 19.88 || 22] 1432800 | 17.97 | .444797 | 18.47 

.376982 | 19.85 || 23} .433878 | 17.93 | .445835 | 18.43 

878123 | 19.83 || 24 | .434954 | 17.92 | .446941.| 18.42 

.879813 | 19.82 ;; 25 | .436029 | 17.68 | .448046 | 18.88 

880502 | 19.78 || 23 | .437102 | 17.87 | .449149 | 18.38 

881689 | 19.75 || 27] .438174 | 17.85 | .450252 | 18.35 

882874 | 19.73 || 28] .439245 | 17.82 | .451353 | 18.32 

.884058 | 19.70 || 29} .440814 | 17.80 | 452452 | 18.32 

885240 | 19.68 |! 8 441382 | 17.78 | .453551 | 18.28 

8.386421 | 19.65 || 81 | 8.442449 | 17.75 8.454648 | 18.95 
.887600 | 19.63 |; 82 | 1448514 |-17.73 | .455748.| 18.25 

883778 | 19.60 || 83} 1444578 | 17.72 | .456838 | 18.22 

.889954 | 19.58 || 84] 445641 | 17.68 | .457931.| 18.20 

.891129 | 19.55 || 25 | .446702 | 17.68 | .459023.| 18.18 

892302 | 19.53 || 86] .447763 | 17.63 | .460114.| 18.15 

893474 | 19.50 || 87 | .448821 | 17.63 | .461203 | 18.13 

.894644 | 19.48 |) 88°] .449879 | 17.62 | .462201 | 18.12 

895813 | 19.45 |; 39 | .4509385 | 17.58 | .463378 | 18.10 

.3896980 | 19.43 |} 40 | .451990 | 17.55 | .464464 | 18.07 

8.398146 | 19.42 || 41 | 8.453048 | 17.55 |8.465548 | 18.05 
.899311 | 19.38 || 42] .454096 | 17.52 | .466631 | 18.03 

.400474 | 19.85 |} 43 | .455147 | 17.48 | .467713 | 18.00 

.401635 | 19.83 || 44] .456196 | 17.48 | .468793 | 18.00 

.402795 | 19.82 || 45 | .457245 | 17.45 | .469873 | 17.97 

.403954 | 19.28 || 46 |. .458292 | 17.438 | .470951 | 17.95 

.405111 | 19.27 || 47 | .459388 | 17.40 | .472028 | 17.92 

.406267 | 19.23 || 48 | .460382 | 17.40 | .473103 | 17.90 

.407421 | 19.22 || 49 | .461426 | 17.37 | .474177 | 17.90 

.408574 | 19.18 |} 50 | .462468 | 17.35 | .475251 | 17.85 

8.409725 | 19.17 || 51 | 8.463509 | 17.32 8.476822 | 17.85 
410875 | 19.13 || 52) .464548 | 17.30 | .477393 | 17.88 

.412023 | 19.13 || 58] .465586 | 17.28 | .478463 | 17.80 

.418171 | 19.08 || 54 | .466623 | 17.27 | .479531 | 17.78 

.414316 | 19.08 || 55 | .467659.| 17.23 | .480598 | 17.77 

415461 | 19.03 || 56 | .468693 | 17.23 | .481664 | 17.73 

.416603 | 19.03 |! 57 | .469727 | 17.20] .482728 | 17.73 

417745 | 19.00 || 58 | .470759 | 17.17 | .483792 | 17.70 

.418885 | 18.98 || 59 | .471789 | 17.17 | .484854 | 17.68 

8.420024 | 18.05 || GO | 8.472819 | 17.13 |8.485915 | 17267 


AND EXTERNAL SECANTS. 


~ 


Vers. 


. | Ex, sec. 


Vers. 


PaielUx. SEC 


COMRIAOUFRWWH OS 


—" 


op oves crew enepenee WIWIWIWIWIWIVISI® Wr : 


| 8.472819 


473847 
474874 
.475900 


476925 
477948 


.478970 
-479991 
481011 
.482029 
483046 


| 8.484062 


.485077 
.486091 
.487103 
.488115 
.489125 


490134. | 


.491141 
.492148 
.493153 
494157 
.495160 
.496162 
497162 
.498162 
.499160 
.500157 


5011538 
502148 | 


. 0038142 


504134 


505125 


.506116 
.507108 


.50§092 |! 


.509079 
.510065 
.511049 
.512033 
.518015 
.513996 


514976 | 


.515955 
.516932 
~517909 


.518684 || 


.519859 


5203832 


.521804 
Sots 


beetle 
523745 
Ov4s714 
020682 


526648 


.527'614 
.528578 
.D29542 
.580504 
.531465 


8.532425 


Ww 09 Cl GO MW Oo 


ie) 


Say AZALI +3 FH MHDMHOO 


Lr 


8.485915 
.486975 
.488033 
.489091 
.490147 
.491202 
.492256 
.493308 
.494360 
.495410 
.496459 
21 8.497507 
498554 
.499600 
.500644 
.501687 
.502730 
.5038771 
.501810 
.505849 
.506887 
. 507923 
.508958 
~5099938 
.511026 
.512057 
.518088 
.514118 
.515146 
.516174 
.517200 
518225. | 
~D19249 


»520272 


.527402 
528416 

529429 
.530441 
.581452 
.532462 
583471 
.pd4478 
580485 
. 536490 
.531495 
.5388498 
.589501 
~)46502 
~541502 

.542501 
.548499 
544497 

545493 


546488 


8.547482 


YAPAZ PAY APY -I-I-V- 


Pm ek ek ek et ek fel fet Pk tp 


> 
fa ee pe ee et ep 
Oe Re eRHEHwtDw wwwsd 


SAPARD a ne - AAAES 


‘yal: 
oO 
cS 


Qn | 


we 


Co 


OF Ee) 


—" 


Peek ped Pd et 
> OUH CO OD mt OOD 


DOWWwWRDWwwowt 


or a Or & OT cy 
FDO SO 


oro 
B= 


8.582425 | 
533384 
. 84342 
.085299 
.586255 
.5387210 
.588163 
.539116 
.540068 
.541018 | 
.541968 
.542916 
.548863 
.544810 


od-4-4 


.545755 
.546699 
.547642 
548584 | 
549525 
.550465 
.551404 


55887 

.559809 

.560738 
8.561666 
. 562592 
.563518 
.064443 
.565367 
.566289 
2567211 
.5681382 
.569052 | 
.569970 | 
.5T0888 
571805 
het 
573636 


579104 | 
580012 
580919 
581825 

582730 

533634 

B3I537 | 
585440 
586341 
STH 
8.538141 | 


ee 


OrOvorgorororvorw orc 


= 
— 


ae) 


Orr ot ur Ui or OF OF UT OU. OT OT Ore 


Fe OTOL. OT OF ST. Gr vor 


8.547482 | 
548474 | 
“549466 
550457 
.BD1447 | 
552436 
553424 
554410 | 
555396 
556381 
557364 
558347 
“559329 
560209 
561289 

562267 
“563245 
564222 
565197 
566172 
567145 
568118 
569090 
570060 
571030 
571999 
572966 
573933 
.574899 
_BT5864 
576827 
577790 


578752 


(0A 


579713 | 
.580673 
.581682 
.582590 
588547 
.584503 
.585458 
.586412 
.5873865 
.588318 
+ 2589269 

.590219 
.591169 
.592117 
598065. | 
,594012 
.594957 
595902 


8.596846 
| 599789 | 
.598731 
599672 
600612 
601551 
602490 

603427 

604363 
97 |B.605299 | 


WWW RIOOWWw 


= 
29) 


> Cd Dd Cd Dd Sd G2 SH? OI S 
CO ODA pe ee Ct OF OT OF 


WITRINVIO WN 


> D> 3 2 > > > > 7D 
hk pak ee DD BA wy 009 G9 09 


wo 


TOCRIDONW OT 


Or =3-2 7 =2- 


> D 
RS) 


| TABLE XXVI.—LOGARITHMIC VERSED SINES 


16° i 

/ Vers. | DA ol Exe sec.) Dei? af Vers. | D. 1’. | Ex. sec.|D. 1’ 
| } | 

0 | 8.588141 | 14.97 | 8.605299 | 15.58 0 | 8.640434 | 14.08 '8.659838 | 14.72 
if .589039 | 14.95 .606234 | 15.55 1 .641279 | 14.07 | .660721 | 14.72 
2 .589936 | 14.95 .607167 | 15.55 2 642123 | 14.05.} .661604 | 14.70 
3 .590833 | 14.93 .608100 | 15.53 3 642966 | 14.05 | .662486 | 14.68 
4 .591729 | 14.90 .609032 | 15.52 4 .648809 | 14.02 | .663367 | 14.68 
5 . 592623 | 14.90 .609963 | 15.50 5 .644650 | 14.02 | .664248 | 14.65 
6 .593517 | 14.88 .610893 | 15.50 6 .645491° | 14.00 | .665127 | 14.65 
a .594410 | 14.87 .611823 | 15.47 7 .646331 | 18.98 | .666006 | 14.63 
8 .595302 | 14.83 612751) 15.45 8 .647170 | 18.97 | .666884 ; 14.62 
: 9 .596192 | 14.83 .613678 | 15.45 9 .648008 | 18.95 | .667761 | 14.60 
iH 10 .597082 | 14.82 .614605 | 15.438 10 .648845 | 18.95 | .668637 | 14.60 
11-| 8.597971 | 14.82 | 8.615531 | 15.42 |] 11 | 8.649682 | 13.93 |8.669513 | 14.58 
12 .598860 | 14.78 .616456 | 15.38 12 .650518 | 13.92 | .6703888 | 14.57 
13 | .599747 | 14.7 .617379 | 15.38 || 13 .651353 | 13.90 | .671262 | 14.55 
14 .600633 | 14.75 .618302 | 15.38 ||} 14 .652187 | 13.88 | .672185 | 14.55 
15 .601518 | 14.75 .619225 | 15.35 15 .653020 | 138.87 | .673008 | 14.52 
Pat (nt | 16 .602403 | 14.72 .620146 | 15.33 || 16 .653852 | 18.87 | .6738879 | 14.52 
Fah 17 .603286 | 14.72 .621066 | 15.33 17 .654684 | 18.85 | .674750 | 14.50 
jaan 13 .604169 | 14.70 .621986 | 15.30 18 6655515 | 138.83 | .675620-| 14.50 
ME 19 .605051 | 14.67 .622904 | 15.380 || 19 .656345 | 138.82 | .676490 | 14.47 
aaa AS .605931 | 14.67 .623822 | 15.28 || 20 .657174 | 13.82 | .677358 | 14.47 
21 | 8.606811 | 14.65 | 8.624739 | 15.27 |) 21 | 8.658003 | 13.78 |8.678226 | 14.45 
Aaa 22 .607690 | 14.63 .625655 | 15.235 || 22 .658839 | 13.7 .679098 | 14.45 
a i 23 .603563 | 14.62 62657 15.23 |} 23 .659657 | 13.77 | .679960 | 14.42 
ta reas .609 445 | 14.60 627484 | 15.23 || 2k .660483 | 18.75 | .680525 ) 14.42 
25; .610321 | 14.60 .628398 | 15.20 || 25 .661308 | 13.7: .681690 || 14.40 
26 | .611197 | 14.57 .629310 | 15.20 || 26 .662132' | 13.73 | .682554 | 14.38 
27 .612071 | 14.57 .630222 | 15.18 27 .662956 | 18.72 | .683417 | 14.38 
23 .6129145 | 14.53 .631183 | 15.17 2 .66377 13.70 | .684280°) 14.35 
29 613317 | 14.53 .639943 | 15.15 29 .664601 | 18.68 | .685141 | 14.35 
i; 30 .614639 | 14.52 .632952 | 15.13 30 .665422 | 13.67 | .686002 | 14.35 
ind 31 | 8.615560 | 14.50 | 8.633360 | 15.13 || 31 | 8.666242 | 13.67 |8.686863 | 14.32 
H 32 .616439 | 14.48 .63£753 | 15.10 || 32 .667032 | 13.65 | .687722 | 14.32 
33 .617299 | 14.47 635674 | 15.10 33 .657881 | 13.63 | .688581 | 14.30 
ed (et| Be! .618167 | 14.45 .636530 | 15.08 || Bt .668699 | 18.62 | .689439 | 14.28 
Bay Wei] 3) .619034 | 14.45 .6374385 | 15.07 35 .669516 | 13.66 | .690296 | 14.28 
i it 33 619991 | 14.42 .638339 | 15.05 35 .670332 | 13.60 | .691153 | 14.25 
a 37 .620765 | 14.42 .639292 | 15.05 || 37 .671148 | 18.58 | .692008 | 14.25 
bin 33 .621631 | 14.40 .640195 | 15.02 || 38 .671953 | 13.57 | .692863°) 14.25 
a 30 .622495 | 14.38 641095 | 15.02 39 672777 | 13.55 | .6938718 | 14.22 
j | 4) .623358 | 14.37 .641997 | 15.00 40 .673590 | 18.55 | .694571 °| 14.22 
Hed 41 | 8.624220 | 14.35 | 8.642897 | 14.98 || 41 | 8.674403 | 13.53 |8.695424 | 14.20 
al 42 .625081 | 14.33 .643796 | 14.97 42 .675215 | 138.52 | .69627 14.18 
13} 43 .625941 | 14.33 644694 | 14.95 43! .676026 | 13.50 | .697127 | 14.18 
NATE 44 .626801 | 14.30 .645591 | 14.95 44 .676836 | 13.48 | .697978 | 14.17 
45 | .627659 14.30 .646488 | 14.93 45 .677645 | 138.48 | .698828 | 14.15 
46 .628517 | 14.28 .647334 | 14.92 || 46 .678454 | 18.47 | .699677 | 14.13 
47 | .629374 14.27 .648279 | 14.90 47 .679262 | 18.45 | .700525 |) 14.13 
43 .6302380 | 14.25 .649173 | 14.88 48 .680069 | 18.43 | .701373 | 14.12 
49 .631085 | 14.23 .650065 | 14.87 49 .680875 | 18.43 | .702220 | 14.10 
50, .631939 ; 14.22 .650958 ; 14.87 50 .681681 ; 18.42 ; .703066 | 14.10 
51 | 8.632792 | 14.22 | 8.651850 | 14.85 || 51 | 8.682486 | 13.40 |8.703912 | 14.07 
52 .633645 | 14.18 .652741 | 14.83 || 52 .683290 | 18.38 | .704756 | 14.07 
53 .634496 | 14.18 -653631 | 14.82 || 53 .684093 | 13.38 | .7O05600°| 14.07 
54 . 6385347 | 14.17 .654520 | 14.80 54 684896 | 13.35 | .706444°! 14.03 
55 .636197 | 14.15 .655408 | 14.80 55 | .685697 | 13.35 | .707286'| 14.03 
56 .637046 | 14.13 656296 | 14.77 56 .686498 | 18.385 | .708128 | 14.02 
57 .637894 | 14.18 .657182 | 14.77 || 57 | .687299.| 138.32 | .708969 | 14.02 
58 .638742 | 14.10 .658068 | 14.7 58 | .688098 | 13.32 | .709810°} 14.00 
59 .639588 | 14.1 10 | .658954 | 14.73 59 .688897 | 13.30 | .710650 | 18.98 
60 | 8.640434 | 14. 08 | 8.659838 | 14.72 |( 49’ 8.689695 | 13.28 '8.711489 | 13.97 


AND EXTERNAL SECANTS. 


COIR erm e | =] 


. | Ex. sec.) D. 1’. 


Ex. sec. 


| 8.689695 
690492 
691289 | 
. 692084 


. 692879 
693674 
694467 
. 695260 
696052 


.696843 | 


697634 
698424 
.699213 
700001 
.700789 
701576 
. 702362 


. 703147 


. 703932 
. 704716 
705499 
. 706282 
. 707063 


107844 
- 708625 
709404 | 


710183 


710961 


711789 
. 712516 
713292 
.714067 


714842 
715616 


. 716389 


717161 
717983 


718704 


719475 | 
720214 | 
721013 


. 721782 


722549 


. 723316 | 
. 724083 
724848 | 


. 725613 
726377 
~ 127140 
.727903 
. 728665 
.'729427 
720187 
730947 


. 731707 


132465 
. 7338223 
. 733981 
134737 
. 735493 


} 8.736248 


woe 3 QV 


(92) 


WWWNWW 9 WW WW WWW WWW 


pose frm poe frm frm fame frre fem ee fe feck Prk peek fomek feed. 


711489 
712327 

713164 | 
“714001 | 
“714838 
115673 
-716508 
717342 
18175 

-719008 

-719840 


420671 
. 721502 
722332 
SOBRE! 
- 723989 | 
724817 
(25644 
726471 
127297 
728122 
728946 
729770 
~730593 
731415 
71382237 
- 738058 
738878 
. 734698 
735517 
. 7363385 
737158 
737970 
738786 
739602 
740417 
7412381 
. 742045 
«742858 
. 748670 
. 744482 


745293 
. 746103 


pea Ate 


on 


OC | 
3.99 


WOIOWIPWWH OS 


739263 


751214 
51955 
752696 
"753436 
54175 


762266 
762998 
763729 
. 764459 
(65189 
.%65918 


. 766647 
67374 
. 768102 
168828 
169554 


Crovrergr grag 


It 
SSSRRGAS 


re 


WOwwnwe 


~ 


eee ee eee 
5 


MDW WO 0b 


“50 


18760578 


.765358 
.766152 
.766946 
167739 
768531 
8.769328 
770114 
770905 
771695 
772484 
173273 


8.785031 
785810 
. 786588 
787366 
788144 
788920 
789696 
790472 
791247 
792021 
8.792795 
793568 
794340 
795112 
795884 
196654 
OTA 
798194 
798963 
199732 
8.800500 
801267 
802034 
802800 
803565 
804330 
805095 
805859 


806622 
8.807385 


~ 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


21° 


Ex, sec. D. 1”. 


COIAMAwWWHS | 


Vers. . | Hx. sec. 4 Vers. Dial 
| 8.780370 5 | 8.807385 | 0 |. 8.822296 | 11.385 |8.852144 | 
.781087 .808147 | 1) Oh .822977 | 11.35 | .852874 
- 781802 .808908 2 .823658 | 11.3838 | .8538604 
782517 .809669 3 .8243838 | 11.38 | 854832 
783231 .810430 4 .825018 | 11.82 | .855061 
783945 .811190 5 .825697 | 11.32 | .855789 
. 784658 .811949 6 .8263873 | 11.80 | .856516 
785371 .812708 7 827054 | 11.28 | .857243 
. 786083 .813466 8 827781 | 11.28 | .857969 
(86794. 814224 9 .828408.| 11.28 | .858695 
- 787505 .814981 10 .829085 | 11.27 | .859420 
788215 8.815737 11 | 8.829761 | 11.25 |8.860145 
788924 .816493 12 .8304386 | 11.25 | .860869 
. 789633 .817249 13 .8381111.| 11.23 | .861593 
- 790342 .818004 14 .881785. | 11.23 | .862816 
-791049 mG .818758 1-15 .8382459 | 11.22 | .863039 
- 791756 Se .819512 16 .8331382 | 11.2 .863761 
792463 3 . 820265 17 .888804 | 11.20 | .864483 
.793169 ai .821018 18 .834476 | 11.20 | .865204 
798874 Be .821770 19 .835148 | 11.18 | .865925 | 
- 794579 we 822521 20 .835819 | 11.17 | .866646 | 
- 795283 .%3 | 8.823272 21 | 8.836489 | 11.17 |8.867365 
. 795987 ory .824023 22 .887159.| 11.17 | .868085. | 
. 796690 ai 824773 23 .887829.| 11.15 | . 868804 | 
(97392 f . 825522 | 24 .888498 | 11.13 | .869522 
798094 826271 | 125 .839166 | 11.13 | .870240 
. 798795 .827019 | | 26 .839834.] 11.12 | .870957 
- 799496 . 827767 27 .840501 | 11.12 | .871674 
.800196 . 828514 28 .841168 | 11.10 | .872390 
. 800896 822261 | 29 .844834 | 11.10 | .873106 
. 801594 .880007 30 .842500. | 11.08 | .878822 
802293 8.830752 | 81 | 8.843165.| 11.07 (8.874537 
.802991 .831497 2 .843829.| 11.07 | .875251 | 
.803688 . 832242 | 38 .844493 | 11.07 | .875965 | 
. 804384 832986 | | 34 .845157 | 11.05 | .876678 | 
.805080 . 833729. | | 85 .845820. | 11.05 | .877391 
80577 . 884472 86 .846483. | 11.03 | .878104 
.806471 .835215 37 .847145. | 11.02 | .878816. | 
.807165 .835957 38 .847806 | 11.02 | .879528 
.807859 .836698 39 .848467. | 11.00 | .880239 
-808552 .837439 40 .849127 | 11.00 | .880949 
8.809244 8.838179 41 | 8.849787 | 11.00 {8.881659 
.809936 .838919 | 42 .850447 | 10.98 | .882369 
.810628 . 839658 43 .851106 | 10.97 | .888078 
.811319 .840396 44 .851764. | 10.97 | .888787 
.812009 .841135 | 45 , 852422. | 10.95 | .884495 
.812699 .841872 || 46 .853079.| 10.95 | .885203 
.815388 . 842609 || 47 .853736. | 10.93 | .885910 
.814077 .843346 | 12. 48 854392 | 10.93 | .886617 | 
.814765 | (844082. ) 12.% 49 .855048 | 10.92 | .887323 | 
.815452 .844817 | 12. 50 .855703. | 10.92 | .888029 
.816139 8.845552 | 12. | 51 | 8.856358 | 10.90 |8.888754 
.816825 .846287 | 12. 11° 52 .857012. | 10.90 | .889489 
817511 .847021."| 12. it 5a .857666 | 10.88 | .890144 
818196 847754 | 12. || 54 .858319 | 10.88 | .890848 
818881 . 848487 | 12. || 55 ~858972 | 10.87 | .891551 | 
.819565 . 849220 | 12.20 || 56 859624 | 10.87 | .892254 
. 820249 .849952 | 12. | 57 .860276.| 10.85 | .892957 
. 820932 .850683 | 12. 58 .860927 | 10.85 | .893659 
.821614 .851414 | 12. 59 .86157 10.83 | .894361 
8.822296 8 852144 | 12. 60 | 8.862228 | 10.82 |8.895062 | 


. . . -_ . ae . . . . . We ~ = 2 é 7 ~ 
WWD CCITT o) I We 5 S ox Ww 0 C9 OV GIR 


CONWWNWWOTO TURN 


BMS oe ap od 5 ag ey oo a 


: ie 
93° 
4 Vers Dig 
0 | 8.862228 | 10.82 
1 .862877 | 10.83 
2 .863527 | 10.80 
3 .864175 | 10.80 
| 4] .864823 | 10.80 
fap | .865471| 10°78); 
6 .866118 | 10.78 | 
7 .86676> | 10.77 | 
8 .86741l | 10.77 | 
9 .868057 | 10.75 | 
10 .868702 | 10.73 
11 | 8.869346 | 10.75 
12 .869991 | 10.72 
13 .870634 | 10.72 
: 14 871277 | 10.72 
15 .871920 | 10.70 
16 .872562 | 10.70 
17 .8738204 | 10.68 
18 .873845 | 10.68 
| 19 .874486 | 10.67 
20 .875126 | 10.67 | 
21 | 8.875766 | 10.65 
22 .876405 | 10.65 | 
| 23 .877044 | 10.63 
24 .877682 | 10.63 
25 | .878320 | 10.62 
26 | 878957 | 10.62 | 
27 879594 | 10.60 | 
| 23 . 880230 | 10.60 | 
29 .880866 | 10.58 | 
| 30] .881501 | 10.58 | 
31 | 8.882136 | 10.58 
32 .882771 | 10.57 
| 33 883405 | 10.55 
34 884038 | 10.55 
35 .884671 | 10.53 | 
36 885303 | 10.53 
| ov .885935 | 10.53 
38 .886567 | 10.52 
39 .887198 | 10.52 
|| 40} .887829.| 10.50 
| 41 | 8.888459 | 10.48 
42 . 889088 | 10.48 
43 | 889717 | 10.48 
44 .890346 | 10.47 
45 .890974 | 10.47 
| 46 .891602 | 10.45 
47 892229 | 10.45 
48 .892856 | 10.43 | 
AS .893482 | 10.43 | 
50 | .894108 | 10.42 
| 
P| 51 | 8.894733 | 10.42 | 
52 | .895358 | 10.42 | 
538. | .895983 | 10.40 | 
54 | .896607 | 10.38 | 
55 .897230 | 10.38 
56 | .897853 | 10.38.} 
57 | .898476 | 10.37 
58 | .899098 | 10.35 | 
59 | .899719 | 10.37 | 
60 | 8.900341 | 10.33 | 


q _ 


AND EXTERNAL SECANTS. 


Taxa sec.) DO 1". i 4 Vers. 
| 
8.895062 | 11.68 || 0 | 8.900841 
.895763 |; 11.67 || 1 | .900961 
896463 | 11.67 |; 2 | .901582 
897163 | 11.65 |} 3 | .902201 
.897862 | 11.65 || 4] .902821 
.893561 | 11.63 | 5'| .903440 
899259 | 11.63 || 6; .904058 
.899957 | 11.63 || 7 . 904676 
.900655 | 11.62 |i § . 905293 
.901352 | 11.60 9 | 905910 | 
.902048 | 11.62 || 10 . 906527 
8.902745 | 11.58 || 11 | 8.907143 
.903440 | 11.60 2 . 907759 
.904136 | 11.57 || 13 . 908374 
.904830 | 11.58 |; 14 . 908989 
pQ0d0eD | ALIS (Wels . 909603 
.906219 | 11.55 || 16 -910217 
.906912 | 11.55 || 17 .910830 
.907605 | 11.53 || 18 .911443 
-908298 | 11.53 |} 19 . 912056 
.908990 | 11.52 || 20 . 912668 
8.909681 | 11.52 || 21 | 8.913279 
-9103872 | 11:52 |} 22 .913891 
911063 | 11.52 || 23 .914501 
917754 | 11:48 || 24 .915111 
.912443 | 11.50 |} 25 915721 
.913133 | 11.48 || 26 . 916331 
.913822 | 11.47 || 27 | .916940 
.914510 | 11.47 || 28 .917548 
:915198 | 11747 |; 29 .918156 
.915888 | 11.45 30 | .918764 
8.916573 | 11.45 |! 31 | 8.919371 
.917260 | 11.48 || 82 .919977 
917946 | 11.43 || 83 | .920584 
.918682 | 11.43 |} 34 . 921190 
.919318 | 11.42 || 335 | .921795. 
.920003 | 11.40 || 383 . 922400 
920687 | 11.42 || 37 .923094 
.9213872 | 11.38 || 88} .923608 
.922055 | 11.40 || 89 | .924212 
.922739 | 11.37 || 40 . 924815 
8.923421 | 11.88 || 41 | 8.925418 
924104 | 11.37 || 42 . 926020 
924786 | 11.35 || 43 . 926622 
.925467 | 11.37 || 44 927224 
926149 | 11.33 || 45 | .927825 
926829 | 11.35 |; 46 | .928425 
927510. | 11.32 || 47 . 929025 
.928189 | 11.83 || 48 . 929625 
928869 | 11.82 || 49 | .930224 
929548 | 11.30 || 50 | .980823 
8.930226 | 11.82 || 51 | 8.931421 
.930905 | 11.28 |} 52 | .932019 
.931582 | 11.30 |) 53 | = .982617 
. 9382260.) 11.27 || 54 933214 
. 932936 | 11.28 || 55 | .933811 
933613 | 11.27 || 56 | .934407 
.934289 | 11.27 || 57 . 935003 
.934965 | 11.25 |; 58 . 935598 
.985640 | 11.25 || 59 . 936193 
8.936315 | 11.23 || 60 | 8.936788 


Dax see. D. 17 
10.33 '8.936815 | 11.23 
10.35 | .936989 | 11.23 
10.32 | .937663 | 11.22 
10.33 | .938336 | 11.22 
10.32 | .939009 | 11.22 
10.30 | .939682 | 11.20 
10.30 | .940354 | 11.20 
10.28 | .941026 | 11.20 
10.28 | .941698 | 11.18 
10.28 | .942369 | 11.17 
10.27 | .943039 | 11.18 
10.27 |8.943710 | 11.15 
10.25 | .944379-| 11.17 
10.25 | .945049 | 11.15 
10.23 | .945718.| 11.13 
10.23 | .946386 | 11.13 
10.22 | .947054 | 11.18 
10.22 | .947722 | 11.12 
10.22 | .948389 | 11.12 
| 10.20 | .949056 | 11.12 
10.18 | .949723 | 11.10 
10.20 |8.950389 | 11.10 
10.17 | .951055 | 11.08 
10.17 | .951720 | 11.08 
10.17 | .952385 | 11.07 
10.17 | .953049 | 11.07 
10.15 | .953713 | 11.07 
10.13 | .954377-| 11.05 
10.13 | .955040 | 11.05 
10.13 | .955703 | 11.05 
10.12 | .956366 | 11.03 
10.10 8.957028 | 11.03 
| 10.12 | .957690 | 11.02 
10.10 | .958351 | 11.02 
10.08 | .959012 | 11.00 
10.08 | .959672 | 11.00 
10.07 | .960332 | 11.00 
10.07 | .960992 | 10.98 
10.07 | .961651 | 10.98 
10.05 | .962310 | 10.98 
10.05 | .962969 | 10.97 
10.03 8.963627 | 10.97 
10.03 | .964285 | 10.95 
10.03 | .964942 | 10.95 
10.02 | .965599 | 10.95 
10.00 | .966256 | 10.98 
10.00 | .966912 | 10.93 
10.00 | .967568 | 10.92 
9.98 | .968223 | 10.92 
9.98 | .968878 | 10.92 
9.97 | .969533 | 10.90 
9.97 18.970187 | 10.90 
9.97 | .970841 | 10.88 
9.95 | .971494 | 10.88 
9.95 | .972147 | 10.88 
9.93 972800 | 10.87 
9.93 | .973452 | 10.87 
9.92 | .974104 | 10.87 
9.92 | .974756 | 10.85 
9.92 | .975407 | 10.85 
9.90 18.976058 | 10.83 


—— 


, Vers. | D: 1". | Ex. sec. | D. 1° / 
0 | 8.936788 | 9.90 | 8.976058 | 10.83 || 0 
1 | .937382 | 9.90] .976708 | 10.83 || 1 
2| .937976 | 9.88) .977358 | 10.83 || 2 
3} 988569 | 9.88 | .978008 | 10.82 3 
4} .939162 | 9.87 | .978657 | 10.82 |} 4 
5 | .989754 | 9.87] .979306 | 10.80 || 5 
6 | .940346 | 9.87 | .979954 | 10.80 6 
7 | .940938 | 9.85] .980602 | 10.80 7 
8 | .941529 | 9.85] .981250 | 10.80 || 8 
9} .942120 | 9.83 - .981898 | 10."8 9 
10 | .942710 | 9.83] .982545 | 10.77 1} 10 
11 | 8.943300 | 9.82 | 8.983191 | 10.77 |] 14 
12 | .943889 | 9.83 | .983837 | 10.77 || 12 
13 | .944479 | 9.80 | .984483 | 10.77 || 13 
14 | .945067 | 9.80 | .985129 | 10.75 || 14 
15.| .945655 | 9.80] .985774 | 10.75 || 15 
16} .946243 | 9.80; .986419 | 10.73 || 16 
17 | .946831 | 9.78} .987063 | 10.7 17 
18 | .947418 | 9.77) .987707 | 10.73 || 18 
19 | .948004 | 9.77 | 988351 | 10.72 || 19 

20 | .948590 | 9.77 | .988994 | 10,72 || 2 
21 | 8.949176 | 9.75 | 8.989637 | 10.70 || 21 
22 | .949761 | 9.75 |} .990279 | 10.72 || 22 
23 | .950346 | 9.75 | .990922 | 10.68 || 23 
24 | (950931 | 9.73 | .991563 | 10.70 || 24 
25} .951515 | 9.73 | .992205 | 10.68 || 25 
26 | .952099 | 9.72 | .992846 | 10.68 || 26 
27 | .952682 | 9.72] .993487 | 10.67 || Q7 
28 | .953265 | 9.70} .994127 | 10.67 || 28 
29 | .958847 | 9.701 .994767 | 10.65 || 29 
30 | .954429 | 9.70 | .995406 | 10.67 || 30 
31 | 8.955011 | 9.68 | 8.996046 | 10.65 || 31 
2} .955592 | 9.68 |} .996685 | 10.63 || 32 
83 | .956173 | 9.67 | .997323 | 10.63 || 33 

34 | .956753 | 9.68] .997961 | 10.63 || 3 
35 | .957334 | 9.65] .998599 | 10.62 || 35 
36 | .957913 | 9.65 | .999236 | 10.62 || 36 
37 | .958492 | 9.65 | 8.999873 | 10.62 || 37 
38 | .959071 | 9.65 | 9.000510 | 10.60 || 38 
89 | .959650 | 9.63} .001146 | 10.62 || 39 
40 | .960228 | 9.62] .001783 | 10.58 || 40 
41 | 8.960805 | 9.62 | 9.002418 | 10.58 || 44 
42 | .9613882 | 9.62] .003053 | 10.58 || 42 
43 | .961959 | 9.60] .003688 | 10.58 || 43 
44 | .962535 | 9.60} .004323 | 10.57 || 44 
45 | .963111 | 9.60 | .004957 | 10.57 || 45 
.963687 | 9.58 | _.005591 | 10.55 || 46 
7 | .964262 | 9.58] .006224 | 10.57 || 47 
48 | .964837 | 9.57 | .006858 | 10.53 || 48 
¢ .965411 | 9.57 | .007490 | 10.55 || 49 
.965985 | 9.57 | .008123 | 10.53 || 50 
| 8.966559 | 9.55 | 9.008755 | 10.53 || 51 
; .967132 | 9.55 | °009887 | 10.52 || 52 
53| .967705 | 9.53] .010018 | 10.52 || 53 
54.| .968277 | 9.53] .010649 | 10.52 || 54 
5 | .968849 | 9.53] .011280 | 10.50 || 55 
.969421 | 9.52 | .011910 | 10.50 || 56 
7 | .969992 | 9.52] .012540 | 10.50 || 57 
.970563 | 9.50 | .013170 | 10.48 || 58 
59 | .971133 | 9.501] .013799 | 10.48 || 59 
0 | 8.971703 | 9.50} 9.014428 | 10.47 || 60 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


Versett) Dt" 
8.971703 | 9.50 |. 9.014428 
. 972273 | 9.48 .015056 
. 972842 | 9.48 .015685 
.973411 | 9.48 .016312 
. 973980 | 9.47 .016940 | 
.974548 | 9.47 . 017567 || 
975116 | 9.45 .018194 | 
.9756838 | 9.45 .018821 | 
. 976250 | 9.43 .019447 
.976816 | 9.43 .020073 
. 9773882 | 9.43 . 020698 | 
8.977948 | 9.43 | 9.021323 
.978514 | 9.42 .021948 
979079 | 9.40 | 022572 | 
.979643 | 9.40 .023197 
.980207 | 9.40 . 023820 
. 98077 9.40 .024444 | 
.981335 | 9.88 | .025067 
.981898 | 9.37 | .025690 | 
. 982460 | 9.38 .026312 | 
. 983023 | 9.37 . 0269384 
8.983585 | 9.35 | 9.027556 
.984146 | 9.35 028177 
984707 | 9.35 .028798 
.985268 | 9.33 . 029419 
. 985828 | 9.33 . 030039 
.986388 | 9.33 .030659 
.986948 | 9.32 .031279 
987507 | 9.32 .031899 
-988066 | 9.32 .032518 
.988625 | 9.30 | .033136 
8.989183 | 9.28 | 9.033755 
.989749 | 9.30 . 034873 
.990298 | 9.28 . 034991 
.990855 | 9.27 . 035608 
.991411 | 9.28 .036225 
.991968 | 9.25 . 0386842 
. 992523 | 9.27 037458 | 
. 993079 | 9.25 .038074 
. 993684 | 9.25 . 038690 
.994189 | 9.23 .039805 
8.994743 | 9.23 | 9.039920 
.995297 | 9.23 040535 
.995851 | 9.22 | .041150 | 
. 996404 | 9.22 .041764 
. 996957 | 9.20 . 042378 
-997509 | 9.22.71 .042991 
. 998062 | 9.18 048604 
.998613 | 9.20 044217 
-999165 | 9.18 . 044830 
8.999716 | 9.17 045442 
9.000266 | 9.18 | 9.046054 
000817 | 9.17 046665 
.001867 | 9 15 047276 
.001916 | 9.17 047887 
.002466 | 9.138 .048498 
0038014 | 9.15 .049108 
.003563 | 9.13 .049718 
.004111 | 9.13 050828 
.004659 | 9.12 . 050937 
9.005206 | 9.12 | 9.051546 


) | 
Ex. see. D. 1’. 


10.47 
10.48 
10.45 
10.47 
10.45 | 
10.45 |} 
10.45 
10.43 
10.43 


| 10.42 


10.42 


10.42 
10.40 
10.42 
10.38 
10.40 
10.38 
10.38 


10.15 


AND EXTERNAL SECANTS. 


/ Vers. | D..1". | Ex, see. |‘D, 1" / Vers, |D.1".| Ex. sec. |D. 1’. 
0 | 9.005206 | 9.12 | 9.051546 | 10.15 || 0 | 9.037401 | 8.77 | 9.087520 | 9.83 
j) -005753 | 9.12) .052155 | 10.13 || 1] .037997 | 8.75 | (088110 | 9/83 
2; .006300 | 9.10} .052763 | 10.13 || 2] .088452 | 8.77 | [088700 | 9.83 
3} .006846 | 9.10 058371, | 10:13 || 3 .038978 | 8.75 .089290 | -9.83 
4| .007392 | 9.10] .053979 | 10.12 || 4] .039503 | 8.73 | 089880 | 9.89 
5) -007988 | 9.08 | .054586 | 10.12 || 5 | .040027.| 8.75 | .090469 | 9.82 
6 | .008483 | 9.08} .055193 | 10.12 || 6] .040552 | 8.73 | 091058 | 989 
7 | ..009028 | 9.07 | .055800.| 10.10 || 7] .041076 | 8.721 091647 | 9/80 
8 | .009572 | 9.07 | .056406 | 10.10 || 8 | .041599 | 8.73 | ‘092985 | 9/80 
9 | .010116 | 9.07 | .057012 | 10.10 || 9} .042123 | 8.72 | 1092823 | 9.80 
10; .010660 | 9.05 | .057618 | 10.10 || 10 | .042646 | 8.70 | .098411 | 9.%8 
11 | 9.011203 | 9.05 | 9.058224 | 10.08 || 11 | 9.043168 | 8.72 | 9.093998 | 9.80 
12 | .011746 | 9.05 | .058829,/ 10.08 || 12 | .043691 | 8.7 094586 | 9.78 
13 | .012289 | 9.03 | .059434 | 10.07 41 13 | .044213 | 8.70 | .095173 | 9.77 
14 | .012831 | 9.03 | .060038 | 10.08 || 14 | .044735 | 8.68 | .095759 | 9.78 
15 | .013373 | 9.03 | .060643 | 10.07 || 15 | .045256 | 8.68] .096346 | 9.77 
16 | .013915 | 9.02 | .061247 | 10.05 || 16 | .045777 | 8.68] .o96932! 9.77 
17 | .014456 | 9.02 | .061850 | 10.07 || 17 | 046298. | 8.67 | 097518 | 9.75 
18} .014997 | 9.02} .062454 | 10.05 || 18 | .046818 | 8.67 | .098103:| 9.77 
19 | .015538 | 9.00 | .063057 | 10.03 || 19 | .047388 | 8.67 | 098689 | 9.75 
20} .016078 | 9.00} .063659 | 10.05 || 20 | .047858 | 8.65 | _og9e74 | 9.78 
21 | 9.016618 | 8.98 | 9.064262 | 10.03 || 21 | 9.048377 | 8.65 | 9.099858 | 9.75 
224 .017157 | 9.00 | .064864 | 10.03 || 22} .048896 | 8.65 | .100443 | 9.73 
Qs 917697 | 8.97 | .065466 | 10.02 || 23! .049415 | 8.63 | 1010287 | 9.73 
24 | .018235 | 8.98 | .066067 | 10.02 || 24 | .049933 | 8.631 .101611 | 9.72 
25 | .018774 | 8.97 | .066668 | 10.02 || 25.| .050451 | 8.63 | 102194 | 9.73 
26 | .019312 | 8.97 |; .067269 | 10.02 || 26 | .050969 | 8.68 | 10277 9.72 
27 | .019850 | 8.95 | .067870 | 10.00 || 27} 051487 | 8.62 | .103361°| 9.70 
28 | .020887 | 8.95 | .068470 | 10.00 || 28} .052004 | 8.60 | .103943| 9.7% 
29 | .020924 | 8.95 | .069070 | 10.00 || 29 | 052520 | 8.62 | 104526 | 9.70 
30 | .021461 | 8.93] .069670 | 9.98 || 30 | .053037 | 8.60 | .105108 9.7% 
31 | 9.021997 | 8.93 | 9.070269 | 9.98 || 31 | 9.053553 | 8 60 | 9.105690 | 9.68 
82 | .022533 | 8.93 | .070868 | 9.98 || 32] .054069 | 8.581 .106271 | 9.7 
33 | .023069 | 8.92 | .071467 | 9.97 || 83] 054584 | 8.58 | 106853 | 9.68 
34 | 1.923604 | 8.92 | .072065 | 9.971; 34 | .055099 | 8.58 | .107434 | 9.68 
35} .024189 | 8.90 | .072663 | 9.97 || 85 | .055614 | 8.58 | .108015 | 9.67 
36 | .024673 | 8.92 | .073261 | 9.97 || 86 | .056129 | 8.57 | .108595 | 9.67 
87 | .025208 | 8.90 | .073859 | 9.95 || 37 | .056643 | 8.57 | .109175 | 9 67 
88} .025742 | 8.88 | .074456 | 9.95 || 38} .057157 | 8.55 | .109755 | 9.67 
3° 026275 | 8.88 | .075053 | 9.93 | 39 | .057670 | 8.55 | .110885 | 9.65 
4) | .026808 | 8.88} .075649 | 9.95 | 40 | .058183 | 8.55 | .110914 | 9.67 
41 | 9.027341 | 8.88 | 9.076246 | 9.93 || 41 | 9.058696 | 8.55 | 9.111494 | 9.63 
42 | .027874 | 8.87 | .076842 | 9.92 || 42} 059909 | 8.53 | .112072 | 9.65 
43 | .028406 | 8.87 | .077437 | 9.93 || 43 | .059721 | 8.53 | 112651 | 9.63 
44; .028938 ; 8.85 | .078033 | 9.92 || 44] .060233 | 8.53 | .113229 | 9.68 
45 | .029469 | 8.85 | .078628 | 9.92 || 45 | .060745 | 8.52 | .113807 | 9.63 
46 | .030000 | 8.85 | .079223 | ‘9.90 | 46] .061256 | 8.52 | .114385 | 9.63 
7 | .030531 | 8.85 | .079817 | 9.92 || 47 | .061767 | 8.50 | .114963 | 9.62 
43 | .031062| 8.83} .080412 | 9.90 || 48 | .062977 | 8.52] 1155401 9.62 
49 | .031592 | 8.83] .081006| 9.88 || 49 | .o62788 | 8.50 | .116117 | 9.60 
50.; 032122 , 8.82} .081599 ; 9.90 || 50; .063298 | 8.48, .116693 | 9.62 
51 | 9.032651 | 8.82 | 9.082193 | 9.88 |) 51 | 9.063807 | 8.50 | 9.117270 | 9.60 
52 | .033180} 8.82| .082786 | 9.87 || 52 | .064317 | 8.48] .117846| 9.60 
53 | .033709 | 8.80] .083378 | 9.88 || 53! 064826 | 8.48] 1184929] 9.58 
54 | .034237 | 8.80] .083971 | 9.87 || 54! .065335 | 8.47 | 118997 | 9/60 
55 | .034765 | 8.80 | .084563 | 9.87 |] 55 | .065843 | §.47 | 119573 | 9.58 
56} .035293 | 8.78 | .085155 | 9.87 || 56 | .066351 | 8.47 | .120148| 9.58 
57 | .035820 | 8.78 | .085747 |. 9.85 || 57 | 066859 | 8.45 | 120798 | 9/57 
58 | .036347 | 8.7 .086338 | 9.85 || 58 | .067366 | 8.47 | .121297 | 9.57 
59 | .036874 | 8.7 -086929 | 9.85 || 59 | 067874 | 8.43 | 121871 | 9.57 
60 | 9.037401 | 8.77 9.83 || 60 8.45 9.57 


9.087520 


9.068380 


9.122445 


417 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


.| Ex. sec. |D. 


/ Vers, | D..1"..). Ex. sec. | D. 1’. 4 Vers. | D. 1’.| ID. 1 
| i | 

0 | 9.068380 8.45 | 9.122445 9.57 0 | 9.098229 | 8.15 | 9.156410 9.30 

1 . 068887 8.43 .1238019 OES (i ekd 098718 | 8.13- . 156968 9.32 

2 .069393 8.43 . 1238593 9.55 2 .099206 | 8.12 | .157527 9.28 

3 . 069899 8.43 . 124166 9.55) |e t3 .099693 | 8.13 | .158084 9.380 

4 .070405 8.42 . 124739 9.53 || 4 .100181 | 8.12 . 158642 9.30 

5 .070910 8.42 .125311 9.55 || 5 .100668 | 8.12 . 159200 9.28 

6 .071415 8.40 . 125884 9.53 6 101155-|- 8.12 159757 9.28 

# .071919 8.42 . 126456 9:53. ||) 7 .101642.} 8.10 . 160314 9.27 

8 .072424 8.40 .127028 9.52 8 .102128.| 8.10 . 160870 9.28 

9 072928 8.40 .127599 9.53 9 .102614 | 8.10 .161427 9.27 

10 078432 8.38 128171 9.52 || 10 .103100 | 8.08 .161983 9.27 

11 | 9.673935 | 8.88 | 9.128742 | 9.52 || 11 | 9.103585 | 8.08 9.162539 ; 9.27 

12 .074438 8.38 . 129313 9.50 || 12 .104070 | 8.08 . 163095 9.25 

13 074941 8.37 . 129883 9.50 +) 13 .104555 | 8.08 . 163650 9.25 

eal 14 075448 8.38 . 130453 9.50 || 14 .105040 | 8.07 . 164205 9.25 
ave it | 15 .075946 §.385 . 181028 9.50 || 15 . 105524 | 8.07 .164760 9.25 
iit 16 .076447 8.37 .1815938 9.50 || 16 .106008 | 8.05 . 165315 9.25 
ea 7 .076949 8.35 .1382163 9.48 7 .106491 | 8.07 . 165870 9.23 
ee Ca 18 .077450 8.35 . 1827832 9.48 || 18 .106975 | 8.05 . 166424 9.23 
ii] a 19 077951 8.35 138301 9.48 |; 19 .107458 | 8.05 .166978 9.23 
Wy 20 078452 8.33 . 1383870 9.47 |\.2 .107941 | 8.08 .167582 9.22 
21 | 9.078952 | 8.33 | 9.134438 9.47 || 21 | 9.108423 | 8.05 | 9.168085 | 9.23 

ile 22 .079452 8.33 . 185006 9.47 || 22 .108906 | 8.03 . 168639 9.22 
th) 23 .079952 8.32 135574 9.47 || 23 .109888 | 8.02 . 169192 9.22 

We hi} 24 .080451 8.32 . 186142 9.45 || 24 .109869 | 8.03 .169745 9.2 
et i 25 080950 8.32 . 186709 9.47 || 25 .110351 | 8.02 .170297 9.22 

Hi 26 .081449 8.32 187277 9.45 || 26 .110882 | 8.02 . 170850 9.20 

2¢ .081948 8.30 . 137844 9.43 || 27 .111313 | 8.00 .171402 9.20 

ty 28 082446 8.30 . 188410 9.45 || 28 .111793 | 8.CO .171954 9.18 
ep | 29 082944 8.28 .1388977 9.43 || 29 .112273 | 8.06 .172505 9.20 
Hh | 30 .083441 8.30 . 1389543 9.43 || 80 .112753 | 8.00 .178057 9.18 
HE 31 | 9.083939 | 8.28 | 9.140109 9.42 || 81 | 9.113233 | 8.00 | 9.173608 | 9.18 
ay Net 32 .084436 8.27 . 140674 9.43 ||: 382 118713 | 7.98 174159 9.18 
33 .084932 8.28 .141240 9.42 || 33 .114192 | 7.98 .174710 9.17 

aa ti} 34 .085429 8227 .141805 9.42 || 34 114671 .| 7.97 .175260 9.17 
patie) 35 .085925 8.25 - 142370 9.40 || 35 .115149 | 7.97 .175810 9.17 
Tuan | 36 .086420 8.27 . 142934 9.42 || 36 115620 - (87 . 176860 9.17 
i 37 .086916 8.25 .1431499 9.40 || 37 .116105 | 7.97 .176910 9.1% 
a | 38 087411 8.25 .144063 9.40 || 88 .116588 | 7.97 .177460 9.15 
bit} 39 .087905 8.23 144627 9.38 || 39 .117061 | 7.95 | .178009 9.15 
40 .088400 8.25 . 145190 9.40 || 40 .117538 | 7.95 .178558 9.15 

uy 41 | 9.088895 8.23 | 9.145754 9.38 || 41 | 9.118015 , 7.93 | 9.179107 9.15 
HG 42\ 089389 | 8.22 | .146317 | 9.38 || 42] .118491 | 7.95 | .179656 | 9.13 
Vea 43 . 089882 8.23 . 146880 9.37 || 48 .118968 | 7.93 .180204 | 9.13} 
44 .090376 8.22 147442 9.38 || 44 119444.) 7.92 180752 | 9.13) 
45 .090869 8.22 .148005 9.37 45 119919 | 7.90 | .181800 9.13 | 
46 .0913862 8.20 . 148567 9.37 || 46 .120895 | 7.92 ; .181848 | 9.12 
7 .091854 8.20 . 149129 9.35 fi 120870 | 7.92! .182895 9.13 

48 .092846 8.20 . 149690 9.35 || 48 121845. | 7.92 | .182943 | 9.12 

49 .092838 8.20 150251 9.37 || 49 .121820 | 7.90 . 183490 9.10 

50 .093330 8.18 . 150813 9.33 |} 50 122294 | 7.90 , .184036 9.12 

51-| 9.093821 | 8.18 | 9.15137 9.35 || 51 | 9.122768 | 7.90 | 9.184583 9.10 

52 .094312 8.18 151934 9.33 1) 52 123242 | 7.88 . 185129 9.10 

53 094803 | 8.17 152494 9.35 || 53 e315 -| ‘7.90 . 185675 9.10 

54 .095293 | 8.17 . 153055 9.32 || 54 .124189 | 7.88 . 186221 9.10 

55 095783 | 8.17 . 152614 9.33 || 55 124662 | 7.87 . 186767 9.08 

56 096273 ; 8.17 .154174 9.32 || 56 .1251384 | 7.88 .187312 9.10 

57 .096763 | 8.15 154733 9.33 || 57 .125607 | 7.87 .187858 | 9.08 

58 097252 | 8.15 | .155293 9.30 || 58 .126079 | 7.87 . 188403 9.07 

59 | .097741 8.13 | .155851 9.32 || 59 .126551 | 7.85 .188947 9.08 

60 | 9.098229 | 8.15 | 9.156410 9.30 || 60 | 9.127022 | 7.87 | 9.189492 9.07 


AND EXTERNAL SECANTS. 


| 9.146126 


| 9.150717 


141971 
142434 
. 142896 
143358 
. 143820 
144282 
144743 
145204 
145665 


146586 
147046 
147506 
147966 
148425 
“148884 | 
"149343 | 
149801 

150259 


~151175 
.151633 
152090 
. 152547 
. 153003 
. 153460 
153916 
154872 | 
9.15 4828 | 


WE AE AY AP APA FAYVAFAY PAF VrFQ VIII INN NNN NNN NN NNN NANN NNNNNVUYNANN NNINQNVNINNVAINN 


CoCow 


IoI-F 


| 9.216971 


. 206799 
.207337 
207874 
. 208410 
208947 
209483 
. 210020 
. 210556 
.211091 


9.211627 
212162 
. 212697 
213232 
213767 
.214301 
. 214836 
-2153870 
215904 
.216437 


217504 
218037 
218570 | 
219102 
219635 
220167 
220699 
221231 

9.221762 


ie 8) 


. 169275 
. 169722 
.170169 
.170616 
. 171062 
Bikes 509 


.175070 
.175514 
175958 
1 Me ne 


.179058 | 


.179500 
. 179942 
. 180388 
.180825 
. 181265 


| 9.181706 


ESS Ay a EE a a NS BHI INN MN NINN NINN 


FY Vers: 2D. 1.) Exec! | D. 1". })°7 Vers. |D.1".| Ex. sec. 
0 | 9.127022 | 7.87 | 9.189492 | 9.07 || 0 | 9.154828 | 7.58 | 9.221762 
1} .127494 85 | .190086 | 9.07 || 1] .155288 58 | .222293 
9| .127965 85 | .190580.; 9.07 || 2 155738 58 | 222825 
3 | .128436 83 | .191124 | 9.07 || 3] .156193 58 | .228355 
4 | .128906 83 | .191668 | 9.05 || 4 | .156648 | 7.57 | .223886 
5 | .129376 83; .192211 | 9.05 || 5! 1157102.) 7.57 | 224417 
6 | "129846 83 | .192754 | 9.05 || 6 157556 | 7.57 | .224947 
7 |. .180316 82} .198297 | 9.05 || ¥ 158010 BY | 225477 
8 | .130785 83 | .193840 | 9.03 || 8 158464 55 | .226007 | 
9 | .181255 82 | .194882 | 9.05 || 9 158917 BB | .226587 | 
10} .131724 80. | .194925 | 9.03 |} 10 15937 55 |. .227066 
11 | 9.132192 80 | 9.195467 | 9.03 |) 11 in 55 | 9.227595 
12 | .132660 82°} .196009 | 9.02 || 12 | .16027¢ 53 | .228125 
18 | .133129 |} 7.784 .196550 | 9.03 || 13 "160728 | 7.53 |. .228653 
14.| .133596 8) | .197092 | 9.02 || 14) .161180 53 | 229182 
15 | .134064 73 | .197633.| 9.02 11 15] .161632 52 | : 229711 
16 | .18453 7 | .198174 | 9.02 || 16 | .162083 53 | .230289 
17 —_ 78 | .198715.| 9.00 |/-17 | .162585 52 | .230767 
18 | .3354( W@ | .199255 | 9.00 || 18 | .162986 52 | .231295 
19 031 ‘7 | .199795 | 9.00 || 19 163437 50 | .231822 
20 | .1363 7% | .200335 | 9.00 || 2 163887 52 |. 282850 
21 | Altea 77 | 9.200875 | 9.00 || 21 | 9.164338 50 | 9.232877 
93 | .137329 | 7.75 | .201415 | 8.98 || 22] .164788 4§ 233404 
23 | 137794 77.| 201954} 9.00 || 93} 165237 50 | .238981 
24 | .138260 73 | .202494 | 8.97 || 24°] .165687 48 | .284458 
95 | .138724 75 | .203032 | 8.98 || 25 |} .166136 48 | .234984 
26 | .139189 43 | .20857 8.98 || 26 | .166585 48 | .235510 
97 | .139653 73 | .204110 | 8.97 || 27 | .167034 48 | .236036 
98 | .140117 3 | .204648 | 8.97-|| 28 | .167488 ” | .286562 
29 | .140581 43 | .205186 | 8.97 || 29 | .167931 7% | .237088 
30 | .141045 72 | .205724 | 8.97 || 80 | .168379 7 | 237613 
31 | 9.141508 2 | 9.206262 95 || 31 | 9.168827 v | 9.238139 


| 9.248602 


238664 
.239189 
.239718 
. 240238 
240762 
. 241286 
. 241810 
. 242333 
242857 
9.243380 
. 243903 
. 244426 
. 244949 
245471 | 
.245994 | 
. 246516 
247088 
.247559 | 
248081 


249123 

249644 | 
250165 | 
250686 | 
251206 | 
251726 

252246 | 
252766 | 


9.253286 


8. 


4 


83 


78 


wJotoFoIssst-g 
NOVI 


CO OV 09 OVOT 


ev) 


~F =F J -JIYF +I 


32° 


Vers. D: 1": 


9.181706 | 7.35 
182147 | 7.33 
.182587 | 7.33 
.188027 | 7.32 
.183466 | 7.33 
.183906 | 7.32 
.184845 | 7.32 
.184784 | 7.32 
185223 ; 7.32 
.185662 | 7.30 


jak 
_ SODIRORWWHOS | ~ 


.186100 | 7.30 
9.186538 | 7.30 

12 | 1186976 | 7.28 
13 | .187413 | 7.30 
14 | 1187851 | 7.28 
15 | .188288 | 7.27 
16 | 188724 | 7.28 
17 | .180161 | 7.27 
18 | .189597 | 7.28 
19 | .190034 | 7.95 
20 | .190469 | 7.27 
21 | 9.190905 | 7.27 
22 | 1191341 | 7.25 
23) 1191776 | 7.25 
24 1192211 | 7.23 
25 | 1192645 | 7.95 
26 | .193080 | 7.28 
27 | 1193514 | 7.98 
28 | .193948 | 7.98 
29 | 1194382 | 7.92 
30 | .194815 | 7.93 
31 | 9.195249 | 7.22 
32 | 195682 | 7.22 
33] 196115 | 7.20 
34] 196547 | 7.22 
35 | .196980 | 7.20 
36 | .197412 | 7.20 
87 | 1197844 | 7.18 
38 | .198275 | 7.20 
39 | .198707 | 7.18 
40 | .199138 | 7.18 
41 | 9.199569 | 7.18 
42 |  .200000 7 
43 | .200430 | 7.18 


44 . 200861 
45 .201291 
46 201720 
47 . 202150 
48 202579 
49 203008 
50 203437 


51 | 9.203866 
52 204294 
53 204723 
54 -205151 
55 205578 
56 . 206006 
57 206433 
58 . 206860 
59 207287 
60 | 9.207714 | 7.10 


bas as doe Se Son ne See es ee Miles Sa tee Des Se ROD LOLS LN 
ms es 
wa or 


Ex. sec. 


9.253286 
253805 
254324 
254843 
255362 
255881 
256399 
256918 
257436 
257954 
258471 
258989 
259506 
260023 
260540 
261057 
26157 
262090 
262606 
263122 
263638 
9.264154 
264669 
265184 
265700 
266214 
266729 
267244 
267758 
268272 
268786 
269300 
269814 
210327 
2710840 
271354 
271866 
272379 
272802 
273404 
213916 
9.274428 
274940 
RUBADR 
275963 
216474 
276986 
277496 
278007 
278518 
279028 


oO 


=) 


| 9.279538 


- 280048 
. 280558 
.281068. 
281577 
282087 
282596 
283105 
.283614 
9.284122 


D. 1’. 


§ 


ot ON x HH DORRRARAAMAAR RAARARRAADAAHD 
SSSISSSSBSE SSSSSSSSSS SSESSRKBSRRARARR 


OO OVO OV oer oor or 
SwWNOwNnwwwe GD G2. G9 MOH GS NON 


OP 20 BCP GD BGO GE.GP 0 G0 GE. Go. G0 G6 G0.GO.|G0 0 60 G0 00 60 G0 40 GD Go GO A 60 GO-GO 66 40.00 60.60.60 G0 00.00 GO GD Go Gd GD G0 00 GD 
NOT OT OTT OT Or ¢ ¢ 


SSmaaonewns| ® 


33° 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


Vers. 


9.207714 
-208140 
208566 
. 208992 
. 209418 
209843 


.210698 
-211118 
.211543 
211967 


9.212391 
212815 
218239 
218662 
214085 
214508 
214931 
215854 


215776 


9.216620 
217042 
217463 
217884 
218305 
-218726 


219567 
219987 
220407 
9.220826 
221246 
221665 
222084 
222503 
222921 
223340 
223758 
224176 


9.225011 
225428 
225845 
226262 
226678 
227095 
227511 
224927 
228342 
228758 
229178 
229588 
230008 
230418 
230832 
231246 
231660 
232074 
232487 


We} 


210268 | 


216198 | 


-219146 | 


224593 


9.232901 


D> D> D> SD D> D2. AE AAT AEAEAPABATAT AE HVAT GPF aP VP gg ge 


Dit 


Qooooeok Hee 
SSRSRRRSSSS 


Noe ye) sO < SO2aooooe eocec|ecocooo 
PODS SESRSERSEEES CO CO C9 OL OT OT OF Ol $3 =F 


2 D2 DISD DAIAHAPAUD ARDAIRMWAAwAOND 
= ° 


Ex. sec. 


9.284122 


-284631 


-285139 


-285647 
- 286155 
286663 
287170 
287678 
. 288185 
. 288692 
289199 


9.289705 
290212 
290718 
291224 
291730 
290236 
292742 
293247 
293753 
294258 
294763 
295268 
295772 
296277 
296781 
297285 
297789 
298293 
298797 
299300 
9.299803 

300807 

800809 

801312 

301815 

802317 

- 802820 

- 808322 

- 803824 

304325 


9.804827 
. 805828 
. 805830 
.806331 
806832 
. 807333 
807833 
. 808334 
808834 
. 809334 


9.309834 
. 3103834 
. 310884 
.311333 
.3811882 
.312331 
.312830 
. 313329 
. 318828 

9.314326 


=) 


1 


e 


APP RL LR RR 


9 OF OF OF 2 OT 3 FIV 


PR oe 
wD Go Go So Go GOH 


WHWUEIWO Wh Wee 
RVIAGIRIGROSIS BHSS 


co 
be 


‘) 


ED ODD OD Oo Go GH Ow Ow Cw ow 
SESSSE SBESRSere 


oo a O0 oy O9 | 
SSEERE 


AND EXTERNAL SECANTS. 


257314 


—————————— aan 


a ee eae eC ae OE a et ee Se hee ets 
| 
34° | 35° 
( 
‘| Vers. | D.1".| Ex.sec.| D.1".|) ’ | Vers. D. 1”.| Ex. sec. |D. 1”. 
0 | 9.232901 6.88 | 9.314326 8.32 0 | 9.257314 | 6.67 | 9.348949 8.15 
1 . 233314 6.88 .814825 CPSU ike .257714 | 6.68 .344438 8.15 
2 . 233727 6.87 -Sloaee. 8.30 9 .258115 | 6.67 344927 8.15 
3} .234139 6.88 .315821 8.380 |} 3 | .258515 | 6.67 845416 8.13 
4 . 234552 6.87 .3816319 8.30 || 4 .258915 | 6.65 .845904 S15 
5 . 234964 6.87 .316817 8.28 5 .259314 | 6.67 .846393 8.13 
6 . 235376 6.87 317314 8.28 6 259714 | 6.65 .346881 S13 
7 .235788 6.85 .317811 8.30 || 7 .260113 | 6.65 .8473869 8.13 
8 .236199 | 6.87 .318309 8.28 | 8 .260512 | 6.65 847857 8.13 
9 . 236611 6.85 .318806 8.28 || 9 .260911 | 6.65 .3848345 8.83. 
10 . 237022 6.85 .319303 8.27 || 10 | .261810 | 6.65 .348833 8.13 
11 | 9.237483 | 6.85 | 9.319799 8.28 || 11 | 9.261709 | 6.63 | 9.349321 8.12 
2 _237844 6.83 . 320296 8.27 || 12 .262107 | 6.63 .3849808 8.12 
13 . 238254 6.85 .820792 §.28 13 .262505 | 6.638 .000295 8.12 
14 | .238665 6.83 .821289 8.27 14 . 262903 | 6.63 . 850782 8.12 
15 . 239075 6.83 .821785 8.27 | 15 .263301 | 6.62 .351269 8.12 
16 .239485 § .82 . 322281 8.25 || 16 .263698 | 6.63 .3851756 8.12 
ve .239894 6.83 .3822776 8.27 17 ‘264096 | 6.62 002243 8.12 
18 240304 6.82 .3823272 8.27 || 18 .2644938 | 6.62 . 852730 8.10 
19 . 240713 6.82 .3823768 8.25 || 19 | .264890 | 6 62 .858216 8.10 
20 | . 241122 6.82 .324263 8.25 2 .265287 | 6.60 .300102 8.10 
21 | 9.241531 6.82 | 9.324758 8.25 || 21 | 9.265683 | 6.62 9.354188 8.10 
92 .241940 6.82 Soe Dee 8.25 || 22 .266080 | 6.60 .354674 8.10 
93 . 212348 6.80 .320748 8.25 || 23 .266476 | 6.60 .3805160 8.10 
24 242756 6.80 .326243 8.23 || 24 . 266872 | 6.58 .305646 8.08 
25 .243164 6.80 .3826737 8.25 || 25 . 267267 | 6.60 .30601381 8.10 
26 .243572 | 6.80 .827232 8.23 26 .267663 | 6.58 .856617 8.08 
27 . 243980 6.7 .3826726 8.23 || 27 .268058 | 6.58 .807102 8.08 
28 244387 6.78 .3828220 8.23 || 28 .268453 | 6.58 007587 8.08 
29 244794 6.78 823714 8.22 || 29 .268848 | 6.58 .3858072 8.08 
30 , 245201 6.7 .3829207 8.23 30 .269243 | 6.58 .808557 8.08 
31 | 9.245608 6.77 | 9.829701 8.23 31 | 9.269638 | 6.57 9.359042 8.07 
32 .246014 | 6.78 .330195 8.22 || 32 | .270032 | 6.57 .859526 8.08 
33 246421 | 6.7 .330688 Sree O38 .270426 | 6.57 .3860011 8.07 
34 . 246827 6.77 .331181 8.22 || 34 .270820 | 6.57 .860495 8.07 
35 . 247233 Gree .33167 8.22 || 85 | .271214 | 6.57 .860979 8.07 
36 .247639 | 6.75 .392167 8.20 || 36 .271608 | 6.55 ,861463 8.07 
37 .248044 | 6.75 .332659 SEO Eas . 272001 | 6.55 .361947 8.07 
38 .248449 6.75 Bs 13 1359 65) 8.20 38 , 272394 | 6.55 .362431 8.05 
39 | .248854 6.75 .333644 8.22 39 272787 | 6.55 .3862914 8.07 
40 | .249259 6.73 .334137 8.2 40 .273180 | 6.538 . 3863398 8.05 
41 | 9.249664 6.73 | 9.324629 8.20 41°| 9.273572 | 6.55 | 9.863881 8.05 
| 42 .250068 Gea fe saDl2t 8.18 || 4: . 273965 | 6.53 .864364 8.05 
43 .250473 6.73 .335612 8.20 || 43 274357 | 6.53 .364847 8.05 
44 .250877 | 6.73 .336104 8.18 || 44 274749 | 6.58 .865330 8.05 
45 .251281 | 6.72 .335595 8.20 || 45 275141 | 6.52 .365813 8.03 
46 .251684 6.7 831087 8.18 || 46 2755382 | 6.53 .866295 8.05 
AT .252088 6.72 .837578 8.18 47 .275924 | 6.52 86677 8.03 
48 .252491 | 6.72 | .338069 8.18 || 48 .276315 | 6.52 .3867260 8.03 
49 . 252894 6.72 . 3838560 8.17 || 4 276706 | 6.52 367742 8.038 
50 | .2538297 | 6.70 . 339050 8.18 || 50 277097 | 6.52 . 868224 8.03 
51 | 9.263699 | 6.72 | 9.839541 8.17 51 | 9.277488 | 6.50 | 9.368706 8.03 
2 .254102 | 6.7 .3840031 8.18 52 277878 | 6.50 .369188 8.03 
53 254504 | 6.70 , 340522 8.17 53 | .278268 | 6.50 .3869670 8.02 
54 . 254905 6.70 .841012 8.17 54 278658 | 6.50 .3870151 8.0% 
55 . 255303 6.68 . 341502 8.15.1) 55 279048 | 6.50 .370632 8 03 
56 .255709 | 6.70 .841991 8.17 | 56 279438 | 6.48 .871114 8.02 
57 .256111 6.68 342481 8.17 BY 279827 | 6.50 011595 8.02 
58 .256512 | 6.68 ,3842971 8.15 58 .280217 | 6.48 . 872076 8.00 
59 .256913 | 6.68 . 843460 8.15 || 59 280606 | 6.48 7 .372556 8.02 
60 : 9. ) 6.67 | 9.348949 §.15 60 | 9.280995 | 6.47 | 9.373037 8.02 


| 


~ 


Vers. 


Ex. sec. 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


Vers. 


o 


. | Ex. sec, | 


| 
| 


ODIDUIP WMHS 


Vo) 


© 


co 


ie) 


9.280995 


. 281883 
251772 
.282160 
. 282548 
. 282936 
. 283324 
.283712 
. 284099 
.284486 
. 284873 
. 285260 
285647 
. 286033 
.286419 
.286805 
.287191 
228757 

5 28) 7962 
. 2883848 
. 288733 
.289118 
. 289502 
289887 
.290271 
. 290655 
. 291039 
. 291423 
. 291807 
. 292190 
.292573 
. 292956 
. 298339 
. 2938722 
. 294104 
. 294486 
, 9948) 68 
2952. 50 
. 295632 

"996014 
. 296395 
296776 
297 157 


297% 538 


7 297 918 
998299 
.298679 

299059 


(299439 


299819 
. 800198 
"300577 
800957 

"301335 
.801714 
. 802093 
302471 


302849 | 

303227 

a 80: 3605 | 
9.303983 


9 OD asi corei mp oie ate a 50D GD ub f 


© 


Jo) 


A AS AES AERTS ERE OO TEE peter SS na 


9.373037 
.0(8018 
.343998 | 
.3814478 
874958 


375438 


375918 
"376398 
376877 


17357 


"377836 
878315 
318094 


379273 


819752 
.880231 
. 380709 
.3881188 
.881666 
302144 
.882622 
.883100 | 
888077 
. 884055 
"384532 
. 885010 
. 885487 
885964 
. 3886441 
886918 


388824 
. 889300 
889776 
.890252 
.3890727 
.3891203 
.3891678 
392154 
.892629 
.3893104 
.893579 
. 394054 
.3894529 
.3895003 
.895478 
.395952 
.896426 
.3896900 | 
8973874 
.3897848 
. 898322 
.898795 
. 399269 
.899742 
400215 
400688 
.401161 
9.401634 


© OF OD OTR WI WHO 


e 
a) 


SSE Shes ee eee ee ee Se See Det he eS Be he Le Ee Eee Eee ere rerers! IAID-FWOMOMOOG 


co 


9.303983 

304360 

304738 

305115 

305492 
305868 
306245 
306621 
306998 
307374 
307749 
308125 
308501 
303876 | 
309251 
309626 
310001 
31037 

310750 
311124 
311498 


.3811872 
.812245 

"312619 
. 312992 

313365 | 
.313738 
.ol4414 
.314484 
.314856 
.315228 


| 9.315600 
815972 | 
816344. 
316716 

317087 | 
817458 | 
817829 
318200 
31857 

318941 


.319311 
.319682 
.820051 
820421 
.820791 
.821160 
. 821530 
.3821899 
2822267 
822636 
. 823005 


823373 


323741 


. 2374 


351409 

ood 77 

O24845 

825212 

as 82! DE 580 

| .825947 
| 9.826314 


So Od G2 GS. S32 S2 Gd G3 Gd Od OentesLae eee 
WWW WWW HWY WW waa 2d 2 


ee ee ee ee a eee 


oO 


© 


Wwwwwwiwe 


9.401634 
.402107 
.402580 
.403052 
.403524 
.403997 
.404469 
.404941 
.405412 
.405884 
.406356 
.406827 
.407298 

407770 
.408241 
.408712 
.409183 
.409653 
.410124 | 
.410594 
.411065 
.4115385 
.412005 
.412475 
.412945 
.413415 
.413854 
.414854 
ae 


15293 


“Ter63 
416281 
.416700 
417168 
417637 
.418106 
418574 
419042 
419511 
.419979 
420447 


| 9. 420915 
.421382 
.421850 
,422317 
.422785 
4 93952 
.423719 
.424186 
.424653 
.425120 
.425587 
.426053 
.426520 
.426985 
.427452 


427918 


.428384 
.428850 
.429316 
9.429782 


Oat IQVANINQ I 


J PEER MEMENTO I a gg gg ng nz 2 
Le ESE : ip PRT Aa hey epee NEG OA sin Dae en len Ch WE ways lI Nee gE ak 


AND EXTERNAL SECANTS. 


89° 


Vers. 


el Eee SOC? 


Vers. 


S) 


ix, sec. D. ale 
) 


i/o) 


We) 


is) 


9.826314 
326681 
827047 
827414 
3827780 
328146 
328512 
3828878 


3829243 | 


.3829609 
829974 


330339 
330704 


.3831069 
331433 


831798 |: 


.332162 
832526 
.332890 
333254 
333617 


.38808981 | 


334344 
8384707 
335070 
335432 
.8380795 


836157 


.836519 


336881 


337248 
837605 
3837966 
838328 


838689 | 


.3839050 
.339411 
83977 
.840132 
340492 
340852 
841212 
841572 


341932 | 
342291 


842651 
843010 
3848369 
84372 

.844086 
844445 
2.44803 
.845161 
845519 
845877 


346235 
346592 
.846950 


.347307 


347664 


9.348021 


9.429782 
430247 
430713 
.431178 
.431643 
432108 
432573 
.433038 
433503 
-433967 
.4384482 


9.434896 
435361 
435825 | 
436289 | 
436753 
437217 
437680 
438144 
-438608 
439071 


9.439534 
.439997 
.440460 
-440923 
.4413886 
.441849 
442312 
442774 
443237 
443699 


| 9.444161 
444623 
445085 
445547 
446009 
446470 
446932 
447393 
447855 
448316 
448777 
449238 
449699 
450160 
450620 
451081 
451541 
452002 
452462 
452922 
| 9.453382 
453842 
454302 | 
454762 
455221 
455681 
456140 
456600 
457059 
9.457518 | 


a3 <3 -3 
| 


OTOTOT OT I 


a3 AF FF I HII 
OMDMIRWUIP WW OS 


C9 OT OD OTOH 


co G2 Ol 


a 2 FoF 


: 3 Fe aPad alas = -5 
WwWwWWNND WWHWwwwe 


a2 J AJrI VWI 


Oe cee Ce ew Lh oe oe he ae te he JINN VIIA Sods Sas Ha aes gs Te ad pt eS 


Jes) 


(Jes) 


| Bap PAB AFA P AP AEAE AE PAPE EEA II 


| 9.348021 
| .348377 
848734 
349090 4 
849446 
849802 
350158 

350514 
350869 
351225 
851580 


9.351935 
352290 
352644 
. 3802999 
. 008000 
.BD38707 | 
.854062 
.804415 
.854769 
.o0123 

9.355476 
.855829 
.856182 
. 856585 
. 856888 
857241 
.8575938 
.857945 
358297 
.858649 

9.359001 
. 8093853 
359704 
. 860056 
.860407 
.860758 
.661108 
.861459 
.861810 
362160 | 


362510 
362860 
. 868219 
.863560 
. 363909 
. 864259 
.864608 
.3864957 
. 865806 
. 865655 
366003 
.866352 
.366700 
.867048 
.3867396 
367744 
.868091 
368439 
.368786 
9 369183 


CRON OT OT OT OT OT ONOTON CLOTOTOTOCOUT OUT OTOH CLOT ON OTOL OT ON OT OU N CLOTOLOT OV OT OT OT OV OT ON 
io 2) 


TOUOT OT 


rovovorercy arororororere 
QD 
i=) 


OU 


e 


orororor 
~ 


Oo 
oon 
wo 


ve) 


), 457518 
457977 
458436 
458895. | 
459355 
459812 
460270 
460729 
461187 
461645 
462108 

9.462561 
463019 
463477 
463934 
464392 
464849 
465307 
465764 
466221 
.466678 


9.467135 
467592 | 
468049 
468506 
468962 
469418 
469875 
470831 
ATN787 
471248 

| 9.471699 

AG 2155 

472611 

473067 

473522 

473978 

474435 

474888 

475343 

475798 

476253 

476708 

477163 

ATT6I8 

48072 

478527 

478981 

479435 

479890 

480344 


.480798 

481252 
481705 
.482159 
482618 
.483066 
483520 
.483973 
484426 
9.484879 


| 


3-3-5 


AE AQ AATF P BF EET AI 


BRAVA 


morgorororororgrcdr ¢ 


OO ort 


UAT AF AQ AR ATAPAT AQP PAP PAPP PIA NIN NINN NINN NNNNNNNNNN 


| 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


40° 


41° 


‘|, Vers. | D. 1". | Ex. see;|.D. 1": 4] 4 | Vers. | D.1".| Ex. sec. |D. 1", 

| eae} a | 

0 | 9.369133 | 5.78 | 9.484879 | 7.55 || 0 | 9.389681 | 5.62 | 9.511901 | 7.45 

1 869480 | 5.78 -4853832 | 7.55 1 .890018 | 5.63 012348 | 7.47 

2 869827 | 5.7 485785 | 7.55 2 890356 | 5.63 | -.512796 | 7.45 

3 BVO174 | 5.77 486238 | 7.55 3 390694 | 5.62 518243 | 7.47 

4 870520 | 5.7 -486691 7.55 4 391031 | 5.62 -513691 | 7.45 

5, .8(0867 | 5.77! .487144 | 7.53 5 3913868 | 5.62 -514138 | 7.45 

6 3871213 | 5.77 |. .487596 | 7.55 6 .891705 | 5.62 -514585 | 7.47 

de .oelpogy BiG -488049 | 7.53 7 -392042 | 5.62 .515033 | 7.45 

8! .371905 | 5.7 .488501 | 7.53 || 8 392379 | 5.62 .515480 | 7.45 

9 872251 |° 5.75 -488953 | 7.55 9} .892716 | 5.60 515927 | 7.45 

10 | .372596 | 5.7 489406 | 7.53 || 10 | .393052 | 5.60 | .516374.| 7.43 

11 | 9.372942 | 5.75 | 9.489858 | 7.53 || 11 | 9.393388 | 5.60 | 9.516820 | 7.45 

12 38738287 | 5.7 -490310 | 7.53 || 12 893724 | 5.62 010267 | 7.45 

13 373632 | 5.75: .490762 | 7.58 || 18 -894061 | 5.58 517714 | 7.48 

14 .873977 | 5.75 | .491214 | 7.52 || 14 -394396 | 5.60 -518160 | 7.45 

15 874822 | 5.75 .491665 | 7.53 || 15 394732 | 5.60 .518607 | 7.43 

oi 16 .374667 | 5.7% 492117 | 7.53 || 16 395068 | 5.58) .519053 | 7.45 
i 17 | .875011 |° 5.75 492569 | 7.52 |) 17 .895403 | 5.58 -519500 | 7.43 
ie 18 | 3875856 | ° 5.7% 493020 | 7.52 |} 18 | .8957388 | 5.60 | .519946 | 7.43 
i 19 -3875700 | 5.7% 493471 | 7.53 || 19 .896074 | 5.58 -5203892 | 7.43 

i 20 876044 | 5.73 493923 | 7.52 || 20] .396409 | 5.57 520838 | 7.43 
21 | 9.376388 | 5.73 | 9.49437 7.52 |] 21 | 9.396743 | 5.58 | 9.521284 | -7.43 
22 8767382 | 5.7% -494825 | 7.52 || 2 897078 | 5.58 .0217380 | 7.43 
Bil 23 377075 | 5.73; .495276 | 7.52 || 23 897413 | 5.57 022176 | 7.42 
| 24 3((419 | 5.72 495727 | 7.52 || 24 897147 | 5.57 022021 | 7.43 
25 307762 | 5.7% -496178 | 7.50 || 25 -898081 | 5.57 | .523067 | 7.43 

fi 26 878105 | 5.'7% 496628 | 7.52 || 26) .3898415 | 5.57] .523513 | 7.42 

i 27 378448 | 5.72 | .497079 | 4.52 || 27 .898749 | 5.57 523958 | 7.43 
28 878791 | 5.7 497530 | 7.50 || 28 899083. | 5.57 .524404 | 7.42 

i 29 | 3791383 | 5.7% 497980 | 7.52 || 29 899417 | 5.55 524849 | 7242 
' 30 | .379476 | 5.7 .4984380 | 7.5 30 | .3899750 | 5.57 | .525294 | 7.42 

| 31 | 9.379818 |. 5.72 | 9.498881 | 7.48 || 81 | 9.400084 | 5.55 | 9.525739 | 7.42 

: 32} .3880161 { 5.7 499331 | 7.52 || 82 | .400417 | 5.55 -526184 | 7.42 
33 880503 | 5.7 .499781 | 7.50 || 33 | .400750 | 5.55 .026629 | 7.42 

34] .380845 | 5.68! .500231 | 7.50 || 34 -401083 | 5.55 -O27074 | 7.42 

35 .881186 | 5.7 .500681 |} 7.50 || 35 .401416 | 5.53 527519 | 7.42 

36 .881528 | 5.68 .501131 7.50 || 36 -401748 | 5.55 .527964 | 7.42 
ca 37 | .381869 | 5.70 | .501581 | 7.48 || 37} .402081 | 5.53] .528409 | 7.40 
38 882211 3.68 50203 7.50 || 388 402413 | 5.53 -528853.| 7.42 
| 3 .882552 | 5.68 502480 | 7.48 || 89] .402745 | 5.53} .529298 | 7.40 
1 40} .882893} 5.68; .502929| 7.50 || 40] 1403077 | 5.5 -529742 | 7.42 
41 | 9.383234 | 5.67 | 9.503379 | 7.48 || 41 | 9.403409 | 5.53 | 9.530187 | ¥.40 

42 | .383574 | 5.68] .503828} 7.48 || 42] 1403741 | 5.53 .530631 | 7.40 

| 43 883915 | 5.67 | .504277 | 7.48 || 43 404073 | 5.52 531075 | 7.40 
tt 384255 | 5.67 504726 | 7.48 || 44 404404 | 5.53 -531519 | 7.40 

45 6884595 | 5.67 .505175 | 7.48 |) 45 | .404736 | 5.52 .931963 | 7.40 

46 .384935 [ 5.67 .505624 | 7.48 || 46 405067 | 5.52 5382407 | 7.40 

7 885275 | 5.67 506073 | 7.48 || 47 | .405398 | 5.52 532851 | 7.40 

48 .885615 | 5.67] .506522 | 7.48 || 48 405729 | 5.50 -533295 | 7.40 
49 | 885955 | 5.65} .506971 | 7.47 || 49 | 406059 | 5.52 | ‘538739 | 28 

50 | .386294 | 5.67 | .507419 | 7.48 || 50 -4063890 | 5.52 534162 | 7.40 

o1 | 9.886634 | 5.65 | 9.507868 | 4.47 || 51 | 9.406721 | 5.50 | 9.534626 | 7.40 

52 .886973 | 5.65 .508316 | 7.48 || 52 407051 | 5.50 535070 | 7.88 

53; 1387312 | 5.65} .508765 | 7.47 || 538 -407381 | 5.50 85513 | 7.33 

54 887651 5.63 509213 | 7.47 || 54 407711 | 5.50 535956 | 7.40 

55 887989 | 5.65 -509661 | 7.47 || 55 | .408041 | 5.50 .536400 | 7.88 

56 888328 | 5.63 -51)109 | 7.47 || 56 | 408371 | 5.4! 536843 7.53 

57 . 3888666 5.65 .510557 Ge YN yay .408700 | 5.50 .537286 7.388 

58 .889005 | 5.68 -511005 | 7.47 || 58 -409030 | 5.48 -537729 | 7.38 

5D | 88938438 | 5.63 51145: 7.47 || 59 .409359 | 5.48 038172 | 7.88 

60 | 9.889681 5.62 | 9.511901 7.45 || 60 | 9.409688 | 5.48 | 9.538615 7.58 


AND EXTERNAL SECANTS. 


412972 


413300 
.413627 
-418955 
.414282 
.414609 
.414936 
415263 
.415589 
415916 
416242 


.416568 
-416894 
.417220 
6417546 
.417871 
.418197 
.418522 
.418848 
419173 
.419498 


9.419822 


-420147 
420471 
420796 
-421120 
.421444 
-421768 
422092 
422416 
422789 


9.423063 


-423586 
-423709 
-424082 
~424355 
424677 


425000 


2425045 
-425967 


| 9.426289 


426611 
426953 
427254 
427576 
427897 
.428218 
428529 


428060 


9.429101 


Ov Or Or orez OW 


oP PR 


aD if 
909 G9 G9 ves 


NGS 
C9 W GI OD 


=o 


OR 
OWW WW 


OV? OVO OT OT ET OT OT OVOTOU OVLOTOV OT OCOT OT OT OU OU OCrorvor ew Ov or ororor or or 
nS 


~) 


~ 


oe 
COwCce 


ree 


oa 


20D wwwowwomwe 


NaI NNW 


OP OT OCOTOUOTOUOT OT GTOTOTOTOLOUOITOTOTOU OLOTOTOTOV OT OVS Or 


oo Ww Go we wows: 
Ce SUE STOUT OT OURS Ot 


Or 


Vers. . | Ex. sec. 
| 9.409688 48 | 9.588615 
| 410017 48 } .539058 
.410346 48 .5389500 
.410675 48 .589943 
.411004 47 .540386 
.411332 47 .540828 
.411660 48 541271 
-411989 47 .541713 
.412317 45 .542155 
.412644 47 542597 
47 


.543040 


9.543482 
548924 
.544366 
.544807 
545249 
.545691 
.546132 
.546574 
.5ATOI5 
.547457 

9.547898 

.548339 

.548781 

549222 

.549663 

.550104 

.550544 

.550985 

551426 

.551867 

.552307 

.552748 

.553188 

.5d58029 

.554069 

.554509 

.554949 

.500889 

.555829 

.556269 

9.556709 

.557149 

.557589 

.558028 

.558468 

.558907 

559847 

.559786 

.560226 

.560665 

.561104 

561548 

.561982 

562421 

.562860 

568299 

.5637388 

256-4176 

.564615 

9.505053 


Jo} 


ve) 


Vers, 


Ex. sec. 


COMOAIHUS WOH © 


— 


aS aQay WEY ATF Y AIPA IF FI IFN VAIN NRA INNS NVRNAANNNNN ANRQRARRRRRER 


WMWWMWWW WWOWe 


QO 


wow 


peo wwwWwIwN w CWO wWtD Ww 


a3 3 3 3-3-7 


—~ 
29 
nH 


9.429181 
.429502 
429822 
-480142 
.430463 
430783 
.431103 
431422 
.431742 
-432062 
482381 

9.432700 
433020 
.433339 
433657 
.433976 
4384295 
.434613 
434982 
485250 
435568 


9.435886 
.436204 
.436521 
.436839 
-437156 
.437473 
437791 
438107 
-438424 
438741 


9.439058 
43937 
-439690 
-440007 
.440328 
-440639 
440954 
-441270 
-441585 
441901 


9.442216 
442531 
442846 
.443161 
.443476 
.443790 
.444105 
444419 
.444733 
-445047 
.445361 
445675 
.445989 
.446302 
446616 
-446929 
447242 
447555 
447868 
9.443181 


ie) 


OOO OTT OC OT OT OT UOT OTT OT, HOT OW OT OWOT UOT UT, OVWOT OWT OE 
WNVWNVYW VWNVVVKVVVAN wz : ¢ 


=) 


CTO OTTO 


| 9.565053 

565492 
.565930 
.566369 
.566807 
567245 
.567683 
.568121 
.568559 
.568997 
.569435 


9.569873 
.570311 
570748 
571186 
.571624 
.572061 
572498 
572936 
573373 
573810 

9.574247 
574685 
575122 
575558 
.575995 
.576482 
.576869 
.577306 


577742 


(((4t0 


578179 
.578615 
~5 19052 
.59 9488 
79924 
.580361 
580797 
.581233 
.581669 
.582105 
.582041 
582977 
.583413 
583848 
.584284 
284720 
.585155 
.585591 
‘586026 
.586462 

.586897 


587332 

58V767 
.588203 
.5886538 
589073 
589508 
589942 
590377 
.590812 
991247 


. . fs 


— 


OOo re rero reir iss, 


a} AT AT AY AF PA PAP AHF EAE PPI I II 


OF AE AF nF nF ag Ba PTF I III NNN NNN NNN 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


44° 45° 
: i 

’ | Vers. | D..1". | Ex. sec. | D. 1’. 4 Vers. 22D, |MEx. see! Dal: 
O | 9.448181 | 5.20 | 9.591247 | 7.23 || 0 | 9.466709 | 5.08 | 9.617224 | 7.20 
1 .448498 | 5.22] .591681-| 7.25 1 .467014.| 5.08! 617656 | 7.18 
2} .448806 | 5.20] .592116-) 7.25 || 2] .467319-| 5.08 |- .61808% | 7.18 
3 | .449118 | 5.22) .592551-| 7.23 3 | .467624./ 5.07 | .618518 | 7.18 
41 .449431 | 5.2 .592985.| 7.23 || 4] .467928 | 5.08} .618949 | 7.18 
5 | .449748 | 5.2 .593419.| 7.25 5 | .468233.| 5.07 | .619380 | 7.18 
6 .450055 | 5.18 .5938854.|  '7.28 6 | .468587./ 5.07 | .619811 | 7.18 
7 | .450366 | 5.20} .594288.' 7,93 7 | 1468841] 5.07 | .620242 | 7.18 
8 | .450678 | 5.20 | .594722.) 7.23 8 | .469145 | 5.07 | .620673 | 7.18 
9 .450990 | 5.18 | .595156.| 7.25 9| .469449 | 5.07 | .621104 | 7.18 
10 | .451201 } 5.18 | .595591 | 7.23 || 10] .469753 | 5.07 | .621535 | 7.18 
11 | 9.451612 | 5.20 | 9.596025 | 7.23 || 11 | 9.470057 | 5.05. 9.621966 | 7.17 
12} .451924 } 5.18] .596459.} 7.23 || 12 | .470360.) 5.07 | .622396 | 7.18 
138} .4522385 | 5.18] .596893.| 7,22 13 | .470664 | 5.05 | .622827 | 7.18 
14. | .452546 |} 5.17] .597826 | 7.23 || 14 | .470967.| 5.05 | .623258 | 7.17 
15 | . .452856 | 5.17 | .597760.| 7.23 | 15 | .471270 | 5.05 | .628688 | 7.18 
16 | .453167 | 5.18 | .598194.| 7.23 || 16 | .471573 | 5.05 | .624119 | 7.17 
7 | ~ 458478) 5.1% .598628. | 7.22 || 17 | .471876 | 5.05 | .624549 | 7.18 
18 | .458788 | 5.17 | .599061.| '7.23° | 18 | .472179 | 5.05 | .624980 |- 7.17 
19} .454098 | 5.17] .599495.| 7.22 || 19 | .472482 | 5.08 | .625410 + “748 
20} .454408 | 5.1% | .599928.| 7.23 || 9 472784. | 5.05 | .625841 | 7.17 
21 | 9.454718 | 5.17 | 9.600862 | 7.22 |} 21 | 9.478087 | 5.03 | 9.626271 | 7.17 
22 | .455028 | 5.17 | 600795 | 7.23 || 22 | - .473889 | 5.08 | .626701 | 7.17 
23 | .455388 | 5.17 | .601229.; 7.22 || 28.) 478691 | 5.03 | .627131 | 7.17 
24} .455648 | 5.15 .601662. | 7.22 |) 24] .473998 | 5.08 | .627561 | 7.17 
25 | .455957 | 5.17 602095 | 7.22 || 25 .474295 | 5.03 | .627991 | 7.17 
26 | .456267 | 5.15 .602528 | 7.23 || 26 474597 | 5.03 | .628421 | 7.17 
27 | 1456576 | 5.15 602962 | 7.22 || 27 | .474899 | 5.02 | 628851 | 7.17 
28 | .456885 | 5.15 603595. | 7.22 || 28) .475200 | 5.03 | .6292981 | 7.17 
29 | .457194 | 5.15 | .603828 | 7.22 || 29 | .4'75502.| 5.02 | .629711 | 7.17 
30.| .457503.| 5.18 604261 | 7.22 || 80 | .475803 | 5.02 | .630141 | 7.17 
31 | 9.457811 | 5.15 | 9.604694 | 7.20 |! 31 | 9.476104 | 5.02 | 9.680571 | 7.17 
32 |- .458120 | 5.15 | 605126 | 7.22 | 82 | .476405 | 5.02 | .631001 | 7.15 
83} .458429-| 5.13 605559 | 7.22 || 83 | .476706 | 5.02 | .631430 | 7.17 
34} .4587387 | 5.13 .605992.| 7.22 || 34 477007 | 5.02.| .631860 | 7.17 
85 | .459045 | 5.13 .606425 | 7.20 || 385 | .477308 | 5.00 | .682290 | 7.15 
36 .459353 | 5.13! .606857 | 7.22 || 36 | .477608 | 5.02 | .682719 | 7.17 
87 | .459661 | 5.18 .607290 | 7.20 || 87 .477909 | 5.00 | .683149 | 7.15 
88 | .459969 | 5.13 607722 | 7.22 || 88 | .478209.| 5.00 | .638578 | 7.17 
3 .460277 | 5.12 .608155 | 7.20 |! 89 | .478509.) 5.00 | .634008 | %.15 
40.| .460584.| 5.13 .608587 | 7.22 || 40 | .478809.| 5.00 | .684437 | 7.15 
41 | 9.460892 | 5.12 | 9.609020 | ‘7.20 || 41 | 9.479109 | 5.00 | 9.684866 | 7.17 
42 | .461199 } 5.12 -609452 | 7.20 |) 42°} .479409.| 5.00] .635296 | 7.15 
43 | .461506 | 5.12] .609884 | 7.20 || 48 | .479709.| 5.00 | .635725 | 7.15 
44 | .461813 | 5.12 .610816 | 7.22 || 44 .480009 | 4.98 | .686154 | 7.15 
45 | .462120 | 5.12 610749 | 7.2 45 | .480308 | 5.00 | .636583 | 7.15 
46 | .462427 | 5.12 -611181.| 7.20 || 46 | .480608 | 4.98 | .687012 | 7.15 
47 | 462734 | 5.10 611613; 7.20+| 47 | .480907 | 4.98 | .637441 | 7.15 
48 | .463040 | 5.12} .612045.|° 7.20 || 48 | .481206 | 4.98 | .637870 | 7.15 
49 | .463347 | 5.10] .612477-) 7.18 || 49 | .481505.| 4.98 | .688299 | 7.15 
50 | .463653 ) 5.10 | .612908 | 7.20 || 50) .481804 , 4.98] .688798 | 7.15 
51 | 9.463059 | 5.10 | 9.613340 | 7.20 || 51 | 9.482103 | 4.97 | 9.639157 | 7.15 
O% | .464265 | 5.10 | .613772.| 7.20 || 52 | 482401 | 4.98 639586 | 7.15 
53 | .464571 | 5.10 | .614204-| 7.18 || 58] .482700 | 4.97 | .640015 | 7.13 
54 | .464877 | 5.10] .614635.| 7.20 || 54 .482998 | 4.97 | .640443 | 7.15 
55 | .465183 | 5.08 | .615067.| 7.20 || 55 | .483296 | 4.98 640872 | 7.15 
56 | .465488 | 5.10-| .615499-| 7.18 |! 56 483595 | 4.97 .641301 | 7.18 
57 | .465794 | 5.08.| .615930.| 7.20 7 | .4888938:] 4.97 | .641729 | 7. 

58 | .466099 | 5.08} .616362.| 7.18 || 58 | .484191 | 4.95 | .642158 | 7. 

59 | .466404 | 5.08 | .616793.} 7.18 |' 59 | .484488 | 4.97] .642586 1 7. 

60 | 9.466709 | 5.08 | 9.617224 1 7.20 | 60]! 9.484786 | 4.97 | 9.643015 | 7. 


S BABAR WWMS | 


Vers. 


AND EXTERNAL SECANTS. 


o 


|/Ex. sec. 


pt 


Vers. 


o 


r ot EX, SCC 


| 9.484786 
.485084 
.485381 


-485678 | 


.485976 
.486273 
.486570 
486866 
487163 


487460 
487756 


9.488053 
.488349 
.483645 
.4889 41 
.489237 
.489533 
. 489828 


.490124 


490419 


.490714 | 


9.491010 
491305 
.491600 
.491894 


-492189 


492484 
492778 


493072 | 


.492367 
.493661 
9.493955 
-494249 


494542 | 
.494836 | 


.495130 
.495423 
.495716 
.496009 
. 496302 
.496595 


| 9.496888 
| 1497181 
497473 
497766 
.498058 


-498350 
493643 


.493935 
.499226 
.499515 
.499810 
pOO101 
.500393 
.500684 
.500975 
.501266 


=) 


501557 


.501848 


502139 


9.502429 


le) 


cers 


| 9.648015 
643443 
643872 
.644300 

.644728 
645156 

645585. | 
.646013 
.§46441 
. 646869 
.647297 
647725 
.648153 
.648581 
.649009 
649436 
.649864 
. 650292 
.650720 
.651147 
.65157 

9.652002 
. 652430 
.652857 
. 6538285 
.6538712 | 
. 654140 
. 654567 
.654994 
.655421 | 
.655849 
.656276 
.656703 
.657130 
.657557 
.657984 
.658411 
.658833 
.659265 
.659691 
.660118 


.660545 
.660972 
.661898 

.661825 
.662252 
.662578 
.663105 
.663531 
.663958 
.664884 
.664810 
665237 
~665863 
6660389 
666515 
666942 
.667368 
667794 
668229 
9.668646 


OOP WmOr © 


1?) 


“I 


fee peek peek peek ek et pt et 
wowuqnwqwoowl! |= 


a as 
WWE OOM 


JIIINIINNT AWAVIANANAN NNVAINABANN NAANINAAANN ANAARAASAAS 


nF AFaJY-I-F-IW-I-I-F 


| 9.502429 


.502720 
.503010. 
. 503300 


.503591 | 


.508881 
504171 
504460 
504750 
505040 
.505329 


9.505618 


.505908 
~506197 
.506486 
506775 
.507063 
.507852 
.507640 
.507929 
.508217 


9.508505 


508793 
.509081 
.509369 
509657 
509945 
. 510232 
.510520 
.510807 
.511094 


9.511381 


.511668 
.511955 
512241 
512528 
512815 
.513101 
.513387 
.513673 
.513959 


9.514245 


.514531 
.514817 
.515102 
.515388 


.515673 | 


.515959 
516244 
.516529 
516814 


| 5 9.517098 
B hoe (BTTS85 


.517668 


.517952 | 


.518236 
518521 
.518805 
.519089 
519378 


9.519657 | « 


PA. A ROR RR RR 
a 
ed 


AtALA AL AAA ALAA AA AAA RADAR AAAS AR RRR ARR 
Mp as eet BBB RIRRINAY : 


W 09 09 0 OH OT OD C9 OT OF 


82 


4.82 


C-~30000 


Roots 


aI FI IIIVIAVINA 
Son! 3 


QIN 


09 OTOTOT SS OF 2 OT 3 


| 9.668646 
669072 
| 669498 
669924 
. 670350 
670776 
671201 
671627 
672053 
672479 
.672904 


9.673330 
673756 
.674181 
674607 
675082 
675458 
675883 
.676809 
676734 
677159 

9.677584 
.678010 
678435 
678860 
679285 
.679710 
.680136 
680561 
. 680986 
.681411 


9.681836 
. 682260 
682685 
.683110 
683535 
.683960 
.684385 
. 684809 
685234 
685659 


9.686083 
686508 
686933 
687357 
687782 
.688206 
688631 
689055 
689479 | 
. 689904. 

| 9.690328 

690752 | 

.691177 

.691691 

692025 

.69 2449 

-692873. | 

693298 
693722 


9.694146 


=z 


Bap ag ay aPaT EE PT I TE TIE SE SS I I I 


AF AB aS aS AS APA TBP 


48° 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


Ex. sec. 


* Vers, »4iD,.1" 1a 

0 | 9.519657 | 4.72 | 9.694146 | 7.07 

519940 | 4.73 | .694570 | 7.07 

2 |» 520224 | 4.72) .694994 | 7.07 

3 | .520507 | 4.73 | .695418 | 7.07 

4/ > 520791 | 4.721 .605842 | 7.07 

D | .521074 | 4.72 -696206 | 7.05 

6 5213857 | 4.72 .696689 | 7.07 

< |. .521640) 4:72 6971138 | 7.07 

8 521923 | 4.72 697537 | 7.07 

9 922206 | 4.70 697961 | 7.07 

10 522488 , 4.72 -698385 | 7.07 
11 | 9.52277 4.72 | 9.698809 | 7.05 
12 | .523054 | 4.7 .699232 | 7.07 
13 | - 523336 | 4.7 .699656 | %.07 
14 523618 | 4.7 .700080 | 7.05 
15 | .523900 | 4.4 .700503 | 7.07 
16 024182 | 4.7 ~TOC92T | 7.05 
17 024464 | 4.7 701850 | 7.07 
18 -024746 | 4.7 TOLTT 7.07 
19 | .525028 | 4.68) .702198 | 7.05 
20 525309 | 4.70 | .702621 | 7.07 
21 | 9.525591 | 4.68 | 9.703045 | 7.05 
22; .525872 | 4.68 103468 | 7.05 
23] 526153 | 4.7 -703891 | 7.07 
24) 526485 | 4.68) .704315 | 7.05 
25} .526716 | 4.68] .704788 | 7.07 
26 526997 | 4.67; .705162 | 7.05 
27 527277 | 4.68 705585 | 7.05 
28 | .52755: 4.68 (06008 | 7.05 
29 | .527839 | 4.67 706431 | 7.07 
30 -528119 | 4.68 706855 | 7.05 
31 | 9.528400 | 4.67 | 9.707278 | 7.05 
32 | .528680 | 4.67 -WO7T7T01 | 7.05 
33 | .528960 | 4.67 708124 | 7.05 
34] .529240 | 4.67 108547 | 7.07 
35 | .529520 | 4.67 -C08971 | 7.05 
36 | .529800 | 4.67 709394 | 7.05 
37 | 5380080 | 4.65 | .709817 | 7.05 
38 -580359 | 4.67 410240 | 7.05 
39 -530639 | 4.65 -710663 | 7.05 
40 -080918 | 4.67 -711086 | 7.05 
41 | 9.531198 | 4.65 | 9.711509 | 7.05 
42 | .5381477 | 4.65 | 711982 | 7.05 
43 | .531756 | 4.65 | .712355 | 7.05 
44} .5820385 | 4.65 712778 | 7.03 
45 | .532314 | 4.63 | .718200 | 7.05 
46 | .5382592 | 4.65) .713623 | 7.05 
47 | 582871 | 4.65} .714046 | 7.05 
48 | .533150 | 4.63 | .714469 | 7.05 
49 | .533428 ) 4.63] .714892 | 7.05 
50, .533706 | 4.65 715815 | 7.08 
51 | 9.533985 | 4.63 | 9.715787 | 7.05 
52 | .5384263 | 4.63 | .716160 | 7.05 
53} .534541 | 4.63] .716583 | 7.03 
54] .534819 | 4.63] .717005 | 7.05 
55 | .585097 | 4.62 | .717428 | 7.05 
56] .585374 | 4.63 | .717851 | 7.08 
57 | .585652 | 4.62 | .718273 | 7.05 
58 | .535929 | 4.63 | .718696 | 7.03 
59 | .5386207 | 4.62 | .719118 | 7.05 
60 | 9.586484 | 4.62 | 9.719541 | 7.05 


CwroomewHe | a 


re 
(or) 


Wt 


-551299 
551571 
-551842 
.5d2118 


552384 


9. 


552656 
552927 


52 


_ 
gr 
oO 


. 742330 
742751 
. 748173 
743595 
744016 
. 744438 
. 744859 


Vers. |D.1”.| Ex. see. 
9.586484 | 4.62 | 9.719541 | 
.586761 | 4.62 ~ 719964 
.587038 | 4.62 . 720886 
.587315 | 4:62 . 726809 
-5387592 | 4.62 . 721231 
.5387869 | 4.60 - 721653 
-588145 | 4.62 . 722076 
.588422 | 4.60 . 122498 
.5388698 | 4.60 722921 
.538974 | 4.62 723343 
.539251 | 4.60 1238765 
9.539527 | 4.60 | 9.724188 
.539803 | 4.60 . 724610 
.540079 | 4.58 725082 
.540854 | 4.60 725454 
.540630 | 4.60 425877 
-540906 | 4.58 - 726299 
.541181 | 4.58 (26721 
.541456 | 4.60 (27148 
-541732 | 4.58 127565 
.542007 | 4.58 727988 
9.542282 | 4.58 | 9.728410 
.542557 | 4.58 . 7288382 
-542882 | 4157 729254 
-543106 | 4.58 . 729676 
.543381 | 4.57 730098 
.543655 | 4.58 . 7380520 
.543930 | 4.57 . 730942 
.544204 | 4.57 . 731364 
.544478 | 4.57 . 731786 
544752 | 4.57 7382208 
9.545026 | 4.57 | 9.732630 
.5453800 | 4.57 . 738052 
1545574) 4257 (88474 
.545848 | 4.55 . 733896 
.546121 | 4.57 . 734817 
.546395 | 4.55 734739 
.546668 | 4.55 . 7385161 
.546941 | 4.55 1385583 
.547214 | 4.55 . 736005 
547487 | 4.55 . 186427 
9.547760 | 4.55 | 9.736848 
.548038 | 4.55 1B 270 
.5483806 | 4.55 . 137692 
548579 | 4.58 .7388114 
.548851 | 4.55 . 738535 
.649124 | 4.53 . 788957 
.549396 | 4.53 ~ 189879 
.549668 | 4.53 . 739800 
.549940 | 4.538 . 740222 
550212 | 4.53 | .740644 | 
9.550484 | 4.53 | 9.741065 
.550756 | 4.538 741487 
.551028 | 4.52 . 741908 
4 he 
4. 
4. 
4. 
4. 
4. 


. 
CN 


OUR IHS | 


Jo) 


AND EXTERNAL SECANTS. 


.5d4009 
.554280 
.554550 
.554820 
.555091 
.555361 
.555631 
.555900 
.556170 
. 556440 
.556709 
.556979 
.557248 
Beta fecn ye 
.557786 
.5D8055 
. 558324 
558593 
. 558862 
.559131 
.559899 
.559667 
.559936 
.560204 
.560472 
.560740 
.561008 
.56127 
.561544 
.561811 
.562079 
. 562346 
.562613 
.562881 
.563148 
.563415 
. 563682 


9.563948 


-564215 
-564482 
-564748 
.565015 
.565281 
.565547 
.565813 
.566079 
.966345 


.566611 
566877 
.567142 
.567408 
567673 
567938 
568204 
568469 
.568734 


9.568999 


D&S 


= 


ee OF 
Co 


So 


LAA LAA RAR A RRR RARER ABRE BRERA BRROROE 
DOAKAIARKIARI’ QBIRVRQRDRWIRHGH BHDDHDOBDSS 


ALR APRA RAR PROD 
9028209 COW Www WwoIW OO 


> 
C29 CO Oe 


S88 


LA a OTC eT’ 


ee 
0 0 


Ex. sec. 


9.744859 
~ 745280 
745702 
-746123 
~ 746545 
. 746966 
. 747388 
747809 
74823 

. 748652 
. 749073 
749494 
.749916 
- 750887 
750758 
751180 
. 751601 
- 752022 
. (52443 
. 752865 
. 1538286 
- 753707 
- 754128 
154549 
154971 
. 155892 
155813 
7156234 
756655 
- 757076 
757498 
757919 
. 758340 
~ 158761 
759182 
759603 
. 760024 
. 7160445 
. 760866 
. 761287 
- 761708 
762129 
762550 
762971 
763392 
- 763813 
- 764234 
. 764655 
. 765076 
. 765497 
. 165918 
. 766339 
. 766760 
167181 
. 767602 
768022 
- 768443 
. 768864 
. 769285 
. 769706 
9.770127 


_ 
S er tenernionas ia | = 


md Ag ag a) oF od og a 


> Ee a J 


wr 


ches be Se oe Se oe MS Ec rs 


z 


~ 


WE AB AE AT AF AF TFA 


| 


| 


Vers. 


9.568999 
569264 
.569528 
569793 
570057 
570322 
-570586 
570850 
571114 
-571878 


571642 


571906 
572170 
972434 
972697 
.972960 
-573224 
573487 
573750 
-974013 
074276 
9.574539 
974802 
575064 
575327 
575589 
575852 
-976114 
.576376 
.5766388 
-576900 


577162 


Vo) 


is) 


O7TT424 | 


577685 
BU947 
578208 
578470 
578731 
578992 
T9253 
519514 
9.579775 
580036 
580297 
580557 
580818 
581078 
581339 
581599 
581859 
582119 


9.582879 


582639" 


582898 
.583158 
.583418 
583677 
583936 
584196 
584455 
9.584714 


Ex. sec. | 


| 9.770127 


770548 
710969 
771389 | 
771810 

72652 
773073 
7 
73014 
714335 


“782330 
82751 
9.78317 
“783592 
784013 
“784433 
“784854 
785215 
“785696 
786116 
"786537 
786958 
9.787378 
787799 
“788220 
“788641 
789061 
89482 
“789908 
“79032: 
790744 
791165 
9.791586 
“792006 
“792427 
“792848 
“793268 
“793689 
“794110 
“794581 
“794951 
9.795372 


SEE EE TEE TE Ty a Ba EE a a aa a ng ng ng ng ng a og 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


~ 


pet 
SODIARTR OTIS | 


© 


© 


Vers. 


9.584714 


.584973 
585232 
.685491 
.585749 
.586008 
586266 
.5E6525 
.586783 
587041 
.587299 


587557 
587815 
.588073 
.588331 
588588 
588846 
.589103 
589861 
.589618 
589875 
590132 
590389 
.590646 
.590903 
.591160 
.591416 
.591673 
591929 
.592185 
592442 


592698 
592954 
593210 
593466 
593721 
593977 
594233 
.594488 
594748 
.594999 


.595204 
.595509 
.595764 
.596019 
.596274 
.596528 
.596783 
.597038 
.597292 
.597546 


597801 | 


598055 
.598309 
598563 
.598817 
599071 
.599324 
.599578 
.599831 
9.600085 


Aw. AR RRO 
39 Go O9 G9 G9 bo G9 by by 
SESH SUND 


Vo) 


Ex. sec. 


~ 


Vers. 


"| Ex. see 


1953872 
2795798 
(96213 
7966384. 
2197055 
T9746 
197896 


.798317 


798788 
799158 
199579 
.800000 
800421 
.800841 
-801262 
.801683 
.802104 
.802524 
802945 
.803366 


803787 


9.804207 
, -804628 


-805049 
.805470 
»805891 
.806311 
806732 
807158 
807574 
«807995 
808415 
.808836 


, +809257 


.809678 
.810099 
.810520 
.810940 
.811361 


 ,811782 


812208 


812624 > 


.818045 


1» .818466 
' (813886 
» 1814307 


814728 
.815149 
.815570 
.815991 


; 816412 


9.816833 
; 1817254 
i .817675 
, .818096 


.818517 


; 1818938 


1819359, 
.819780 
820201 


9 -820622 


=) 
vo 
{ 


i=) 


SOWWW 


We 


Oo 


COOIAIOUP WMH © 


Se Scoocessooe 
_ 


WD WW 


S 
OWS 


AE AE AT AF AE ATVB ATA AE I NINN NNN NNNNNNNNNN NNNNNNNNNS NNN ANARANN 


9.600085. | 


.600338 
.600591 
. 600845 
.601098 
.6013851 
- 601603 
. 601856 
. 602109 
. 602362 
. 602614 


| 9.602866 


.603119 
.60337 
. 603623 
603875 
-604127 
.604879 
-604631 
.604883 
.605134 


9.605386 


- 605637 
.605888 
-606140 
.606391 
-606642 
. 606893 
.607144 
607394 
- 607645 


9.607896 


.608146 
.608397 
.608647 
608897 
.609147 
. 609397 
.609647 
.609897 
.610147 


9.610397 


.610646 
.610896 
.611145 
.611394 
.611644 
.611893 
.612142 
.612891 
.612640 


9.612&88 


.613187 
. 613386 
.613634 
.613883 
.614131 


.614379 | 


. 614627 
614876 


| 9.615124 


AAA AA LAA OA RAR AAR ALAA AAR ALR ALAA PAAA ARASH A PAPERS SEAA PREPRESS 
Re d+ 2 2D Ww dD? > wwMNHNNNNH NW? 


— 
— 
eo GO 


| 9.820622 
. 821043 
.821464 
. 821885 
. 822306 
822727 
.823148 
. 823569 
. 823990 
824411 
. 824833 

9.825254 

.825675 

. 826096 

.826517 

. 826938 

.827360 

827781 

. 828202 - 

. 828623 

.829044 

. 829466 

829887 

. 830308 

. 830729 

.831151 

.831572 

.831993 

.8382415 

832836 

.888257 

. 883679 

. 834100 

. 884522 

. 8384943 

835364 

. 835786 

.836207 

. 836629 

.887050 

8387472 

.837893 

.8388315 

. 838736 

.8389158 

.8389579 

.840001 

, 840423 

. 840844 

.841266 

.841687 

9.842109 
. 842531 
.842953 
. 843374 
. 843796 
.844218 
.844639 
. 845061 
. 845483 

9.845905 


AUNT AE APAYAT AT VATA AIA NII NINA ANAND NAN NN NNNNIN NNNNNNNANN RANNRANARAERR 


Jat =3 


~ 


wre | 


oo, 


Ct 


ocosto 


10 


c 


iJon) 


© 


ie) 


.615619 
.615867 
.616115 
.616362 
.616610 
.616857 
.617104 
.617551 
.617599 
617845 
.618092 
.618339 
.618586 
. 618833 
.619079 
.619326 
.619572 
.619818 
.620065 


620311 
.620557 
620803 
621048 
.621294 
621540 
621786 
622031 


622276 


622522 


~URRVIKRA 
. 622767 
.623012 
. 623257 
. 623502 
623747 
. 6238992 
. 624237 
. 624481 
. 624726 
.624970 
.625215 
.625459 
.625703 
.625947 
. 626191 
.626435 
. 626679 
. 626923 
.627166 
.627410 
. 627654 
.627897 
.628140 
. 628384 
. 628627 
.628870 
.629113 
. 629356 
629598 


9.629841 


AND EXTERNAL SECANTS. 


9 


Ex. sec. 


~ 


Vers. 


Ex. sec. | 


ASA AAAS AAS PRA AAA ALAA ALAR AAA ADA PARA RAR RR 


4 | 
( 


9.845905 


840527 
.846749 
847170 
847592 
848014 
848436 
.848858 
849280 
.849702 
850124 


9.850546 
850968 
851390 
851812 
852234 
852656 
853078 
853500 
853923 
854345 

9.854767 
855189 
855612 
856034 
856456 
856878 
857301 
857723 
858145 
858568 

9.858990 
859413 
859835 
860258 
. 860689 
861103 
861525 
.861948 
862370 
862793 

9 863215 
.863638 
.864061 
.864483 
864906 
865329 
865752 
866174 
866597 
867020 

9.867443 
867866 
868289 
868712 
.869135 
869558 
869981 
870404 
870827 

9.871250 


COOVIAOoR WMH © 


=) 


ba en Sens Sous Sone Sen Dent Deas ae Se Silas Ses es Se De hh Le Ss hh Lt oR ae en Pa ce ce ee ee eee: 


9.629841 
.639084 
. 630326 
. 680569 
. 630811 
.631054 
.631296 
.63153 
.631780 
.682022 
. 682264 

9.632505 

682747 

. 632989 

.633230 

.683472 

.633713 

. 6388954 

. 634196 

.6384437 

.634678 

.684919 

.685159 

.6385400 

.685641 

.685881 

.636122 

. 636862 

.686603 

686843 

.637083 

9.637323 
.687563 
.687803 
.638043 
. 638283 
.688522 
. 638762 
.639001 
.689241 
. 639480 


6389719 
. 689958 
.640197 
. 640486 
.640675 
.640914 
.641153 
.641891 
.641630 
.641868 
642107 
642345 , 
. 642583 
642522 
.643060 
648298 
.648585 
64877 
.644011 
9.644249 


SoS ooocoodoeo°ococo 


SCwwo SSSSRBBSESES CO 09 CO 09 CO 09 OT 09 OLN Or 


Seooeoocose soooSesso 


Sime 


=) 
oS 


Sooo > 
CoOooo CO: 


le) 
@ 


MCPD COCCI CDCI COCICIDWNWNWW WOK AOD AAA AAD ARAL AAA PAD ADALE AA ARAAAREA ABE 


9.87125 
.871673 
.872096 
872519 
-872942 
.873266 
.873789 
874212 
.874636 
.875059 
875482 
9.875906 
.876829 
.876752 
877176 
.877599 
.878023 
.878446 
878870 
879294. 
879717 
9.880141 
.880565 
-880958 
.881412 
.881885 
.882260 
- 882583 
-883107 
.888531 
.883955 
9.884379 
88480 
-885227 
.885651 
.886075 
.886499 
. 886923 
887847 
887772 
.888196 
9.888620 
889044 
.889469 
.889293 
.890317 
.890742 
.891166 
.891591 
.892015 
892440 


9.892864 
893289 
893714 
.894188 
894563 
894958 
.895412 
895087 
.896262 

9.896687 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


_ 
owome- 


OOP WMO 


Vers Hix, secy| 1 1".71) <{ Vers... | D. 1".|, Ex. sec..|D. 1° 
| 

| 9.644249 | 3.95 | 9.896687 | 7.08 0 | 9.658356 | 3.87 | 9.922217 | 7.18 
644456 | 3.9% 897112 | 7.08 1 658588 | 3.88 922074 | 7.13 
644724 | 38.95 897537 | 7.08 2 .658821 | 3.87 .923102 | 7.12 
644961 3.95 897962 | 7.08 3 .659053 | 3.88 928529 7.12 
"645198 | 3.95 | .898387 | 7.08 || 4 | .659286 | 38.8% | .920056 | 7.135 
"645435 | 3.97 | .898812 | 7.08 || 5 | .659518 | 8.87 | .924e84 ) 7.18 
645673 | 3.95 | .899237 | 7.08 || 6 | .G59750 | 8.88} .924511 | 7.19 
645910 | 3.95 | .899662 | 7.08 || 7% | .659083 | 8.87 | 925209 | 7.12 
"46147 | 3.95 | .900087 | 7.08 || 8] .660215 | 8.87 | .9256C6 | 7.13 
646884 | 3.93 .900512 | 7.10 9 660447 | 3.87 926094 | 7.12 
"646620 | 3.95 | .900988 | 7.08 || 10 | .660679 | 38.85 | .926521 | 7.13 
9.646857 | 3.95 | 9.901863 | 7.08 || 11 | 9.660910 | 3.87 | 9.926049 | 7.13 
"47004 | 3.93 | .901788 | 7.08 || 12 | .661142 | 3.87 | .920877 | 7.12 
.647330 | 3.95 .902213 | 7.10 |} 13 .661874 | 3.85 927804 | 7.13 
647567 | 3.98 902689 | 7.08 || 14 .661605 | 3.87 928232 | 7.13 
"647803 | 8.93 | 902064 | 7.10 [15 | .661837 | 3.85 | .928660 | 7.138 
"648039 | 3.95 | .903490 | 7.08 || 16 | .662068 | 8.87 | .920088 | 7.13 
"648276 | 3.93 | .903915 | 7.10 || 17] .662500 | 8.85 | .929516 | 7.138 
.648512 | 3.93 904341 7.08 || 18 .662581 | 3.65 .929944 | 7.18 
.648748 | 3.93 904766 | 7.10 || 19 .662762 | 3.85 930372 | 7.18 
"648984 | 3.93 | .905192 | 7.08 || 20 | .662993 | 3.85 | .930800.) 7.18 
9.64922 3.93 | 9.905617 | 7.10 |] 21 | 9.663224 | 3.85 | 9.931228 | 7.13 
"49456 | 3.92 | .906043 | 7.10 || 22 | .663455 | 8.85 | .981056 | 7.15 
649691 3.93 .906469 | 7.08 || 2 663666 | 8.85 982085 | 7.13 
.649927 | 8.93 | .906804 | 7.10 || 2 663917 | 8.85 | .982513 | 7.13 
.650163 | 38.92 907820 | 7.10 || 25 664148 | 3.83 982941 7.13 
"650398 | 8.92 | .907746 | 7.10 || 26] .664378 | 8.85 | .988869 7) 7.15 
650633 | 3.93 | .908172 | 7.10 || 27 | .664609 | 8.83 | .983708 | 7.13 
"50869 | 3.92 | .908598 | 7.10 || 28] .664889 | 3.85 | .984226 | 7.15 
.651104 | 3.92 .909024 | 7.10 || 2 .665070 | 38.83 . 984655 7.13 
651339 | 3.92) .909450 | 7.10 || 80 | .665300 | 3.83 | .980033 | 7.15 
9.651574 | 3.92 | 9.909876 | 7.10 || 81 | 9.665530 | 8.83 | 9.935512 | 7.15 
.651809 | 3.92 .910802 | 7.10 || 382 .665760 | 8.83 935941 7.13 
.652044 | 3.92 910728 | 7.10 || 33 .665990 | 3.88 9803869] 4.15 
652279 | 3.92 .911154 | 7.10 || 34 666220 | 8.88 9386798 | 7.15 
652514 | 3.90 911580 | 7.10 || 35 .666450 | 3.83 9387227") 7.15 
652748 | 3.92 .912006 | 7.10 || 36 .666680 | 3.83 .987656°| 7.15 
.652983 | 3.90 912482 | 7.12 || 37 .666910 | 8.82 9388085 | 7.138 
.650217 | 3.92 912859 | 7.10 || 88 | .667139 | 3.83 .988513°| 7.15 
"653452 | 3.90 | .918285 | 7.10 || 89 | .667369 | 8.83 | .988042°| 7.15 
"653686 | 3.90 | .913711 | 7.12 || 40 | .667599 | 8.82 | .989871°) 7.1% 
9.653920 | 3.92 | 9.914188 | 7.10 || 41 | 9.667828 | 3.82 | 9.939801 | 7.15 
.654155 | 3.90 914564 | 7.12 || 42 .668057 | 3.83 940250") 7.15 
.654889 | 3.90 914901 7.10 || 43 66287 | 3.82 .940659°} 7.15 
654623 | 8.90 | .915417 | 7.12 || 44 | 668516 | 3.82 | .941088 7.15 
654857 | 3.88 915644 | 7.10 || 45 668745 | 3.82 OATON Ts le Ele Le 
‘655090 | 3.90 | .91G270 | 7.12 |] 46 | .668074 | 3.82 | .941947 | 7.15 
655824 | 3.90 | .916G07 | 7.12 || 47 | .660203 | 3.82 | .9423876 | 7.15 
.655558 | 3.90 917124 | 7.10 || 48 | .669482 | 3.82 .942806.| 7.15 
.655792 | 3.88 917550 | 7.12 || 49 | .669661 | 3.80 .948285°| 7.17 
(656025 | 3.88] .917977 | 7.12 || 50 | .669889 | 3.82 | 948665.) 7.16 
9.656258 | 3. 7.12 || 51 | 9.670118 | 3.82 | 9.944094 | 7.17 
656492 | 3.8 7.12 || 52 | .670847 | 8.80 | .944524| 7.15 
. 65672 3. 7.12 || 53 | .G70575 | 8.82 | .944953 | 7.17 
.656958 | 3.8 7.12 || 54 | .670804 | 8.80 | .945383'| 7.17 
657191 | 38. 7.12 || 55 | .671082 | 8.80 | .945813°) 7.17 
657424 | 3.88 920589 | 7.12 || 56 .671260 | 3.80 .946243°| 7.17 
:657657 | 8.88] .920966 | 7.12 || 57 | .671488 | 8.80 | .946673 7.17 
657890 | 3.88 | .921303 | 7.12 || 58} .671716 | 8.82 | .947103 7.17 
.658123 -€8 | .921620 | 7.12 |) 59 | 671945 | 3.78 | .947583°| 7.17 
9.658356 | 3.87 | 9.922947 1 7.12 || 60 | 9.672172 | 8.80 | 9.947063! 7.1% 


»AND EXTERNAL SECANTS. 


o re 


Ne) 


Vers. 


| 9.672172 


. 672400 
.672623 
. 672856 
.673083 
.673311 
.673533 
.673766 
.673993 
67422 

674448 
.674675 
.674902 
.675129 
.675356 
675582 
. 675809 
. 676036 
676262 
.676489 
.676715 
676941 
.677168 
.677394 
. 677620 
.677846 
.678072 
.678298 
.678523 
.678749 
.678975 
.679200 
.679426 
.679651 
.679876 
. 680102 
. 680327 
. 680552 
.680777 
.681002 
.681227 


9.681451 


.681676 
.681901 
682125 
. 682350 
682574 
682798 
. 683023 
683247 
683471 
683695 
683919 


.684143 | 


. 684367 
. 654590 
684814 
.685037 
68561 
.685484 


| 9.685708 


Ex, sec. 


Vers. 


| Jox. sec. 


CERN TIN NIS 


a ate FI 


32 3 2 III 


© OTOT OT OT OT =2 OF OT 


~3 
OU 


wWecew$c oc cotcec wow tec cote Coco 


IWAN 


tS We OO COW OO 


9.947963 
. 948393 
. 948823 
. 949253 
. 949683 
.950114 
. 950544 
. 950975 
. 951405 
. 951836 
952266 
. 952697 
. 953128 
.953558 
.9538989 
. 954420 
.954851 
. 955282 
.9355713 
956144 
. 956575 
9.957006 
.957438 
. 957869 
. 958300 
. 958732 
.959163 
.959595 
. 960026 
. 960458 
. 960890 
9.961321 
.961753 
.962185 
. 962617 
. 963049 
. 963481 
.963913 
. 964345 
96477 
. 965210 
9.965642 
. 966075 
. 966507 
. 966940 
. 967372 
. 967805 
. 968238 
968670 
. 969103 
. 969536 


. 969969 
. 970402 
. 9708385 
. 971268 
.971701 
. 9721385 
972568 
973001 
. 973435 
9.973868 


o 


eo) 


WOONROUR WOH OS 


AP AE AT AT AQ APPT IVI AI III - 


II RQ WII 


| 9.685705 | 


. 685931 


.686154 | 


.686377 
. 686800 
. 686823 
.687046 


687269 


. 687492 
687714 
687937 
9.688159 
. 688382 
688604 
-683826 
. 689048 
689271 
689493 
.689715 
-689937 
.690158 


9.690380 
. 690602 
. 690823 
-691045 
.691266 
.691488 
.691709 
.691930 
.692151 
692372 

9.692593 
.692814 
.693035 
.693256 
693477 
.693697 
-695918 
.694138 
.694359 
.694579 


694799 
.695019 
.695240 
.695460 
.695680 
. 695899 
.696119 


=) 


. 696339 | 


. 696559 
.696778 


.696998 
.697217 
. 697437 
.697656 
697875 
698094 
698313 
698532 
698751 


Je) 


9.698970 


co oo G9 G9 G9 G9 CD C9 OD OD OD 


JIA VII MIM MMII 
SWOwWWNwwnwwey 


TOO OWooOew 


Sor eS Bo 0) Bel lie en a 
a 


co 9 9 Go C8 G9 CO OF G9 CO CO 


w) 
“ 


509 © 


Wwiwewwtcoct cwtccocmc NOI ce co cu oN G9 Co cu to GY OO CO 
3 Fe SS SO SAE 2 FAAARPADMH >HI AAIDP OOS 


| 9.973868 


974802 
974736 
. 975169 
975603 
. 976037 
976471 
976905 
97733 


ITT 


978207 


9.978641 


1979075 
.979510 
. 979944 
. 980879 
. 980813 
. 981248 
- 981682 
982117 
982552 


9.982987 


. 983422 
. 983857 
- 984292 
. 984727 
. 985162 
. 985597 
- 986033 
. 986468 
. 986904 


9.987339 


lpdedou-4 
: 987 (40 


. 988210 
. 988646 
. 989082 
. 989518 
. 989954 
. 990390 
. 990826 
991262 


9.991698 


. 992134 
99257 


. 993007 


. 9938444 


. 9983880 
.994317 
. 994754 
. 995191 
995627 


9.996064. 
996501 
. 996938 
. 997376 
997813 
. 998250 
. 998687 
999125 

9.999562 


10.000000 


WP AE ABAD PATE A aaa INI INNIS NHN NNNINNNN NNNNNNAANEN 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


| 


oumcswee | 


mWMwre COONS 


Vers. 
9.698970 
. 699189 
.699407 | 
.699626 
. 699845 
100063 
700282 
-700500 
. 700718 
. 700936 
-701154 
701372 
701590 
.701808 
. 702026 
. 702244 
702462 
102679 
~T02897 
. 703114 
. 703832 
. 703549 
. 703766 
. 703983 
704200 
- 704417 
. 704634 
704851 
. 705068 
05285 
. 705501 
. 705718 
- 105935 
. 706151 
106367 
. 106584 
. 706800 
. 707016 
- (OF282, 
107448 
. 707664 
. 707880 
. 708096 
. 708811 
708527 
. 7087438 
08958 
WO91LT4A 
.709389 
.709604 
. 709819 
. 710085 
.710250 
. 710465 
.710680 
.710895 
~711109 
. (113824 
~711589 
111758 


9.711968 


NNO MMM AMARe SRRADIPOSOS CONS SSSS6O SONS SOSSSS Sane Sees: 


Doo Go 
Co OU 


Ex. sec. 


| 10.000000 


.0004388 
000875 
.001313 
001751 
.002189 
.002627 
. 008065 
. 003503 
.003942 
.004380 


10.004818 
005257 
.005695 
.006184 
.006573 
.007012 
. 007450 
007889 
.0083828 
008767 

10 .009207 
.009646 
.010085 
.010525 
.010964 
.011404 
.011843 
.012283 
012723 
.01381638 

10.013603 
.014043 
.014483 
.014923 
.015363 
.015804 
.016244 
.016684 
017125 
-017566 

10.018007 
.018447 
.018888 
.0193829 
.019770 
-020212 
.020653 
.021094 
.021535 
.021977 
022419 
. 022860 
023302 
023744. 
024186 
. 024628 
. 025070 
:025512 
025954 

10. 026297 


aomewne | 


— 
THe WWM OOO 


TEE ENE NEE a a a gt ER HEE 


9 


Vers. 


- 711968 
712182 
712397 
712611 
712825 
.718039 
718253 
- 713467 
718681 


718895 


.718870 
718582 
. 18794 
.719007 
. 719219 
.719431 
719643 
719855 
. 720066 


720278 , 


720490 
- 720701 


Bit 
; 
9.0 


S 


.| Ex. see, 


ew 


SP LOSE CORO CO COO COED CY CD CIDDEDCD OI ODCIOD CUCUCCDEDEDEDEDEODED CIDCCUUDOIODG WeDeDCcDEceDecD cE EOI ECD OD CD ON OH OD 
: mare OS at 5 EO Ee aca Ce ese CO COLON ET OC EY Oh Oh Oe a ae a Sr ait Gr SSF SS ae Dv ee ee 


1 


10.026397 
026839 
.027281 


ORT724 


028167 


028609 
.029052 
.029495 
029938 
.030381 
030825 
.031268 


031711 


.032155 
. 032598 
.033042 
.033486 


.033929 | 


034373 
.034817 
.035261 
10.035705 
-036150 
-936594 
037038 
.037483 
037928 
038372 
. 038817 
039262 
089707 


10.040152 
040597 
.041042 
-041488 
.041933 
042379 
042824 
. 043270 
043716 
.044162 

10.044608 
045054 
.045500 
045946 
- 046393 
046839 
047286 
047732 
.048179 
. 048626 

».049073 
049520 
049967 
050414 
050861 
.051309 
051756 
052204 

052652 


10.053099 


pas oma, as Dent ake Sr ras Si as De dons Sons Ss So Sons Ss Se ln Son ous Sow Sows Bs Ss Sow De Ses ies Se De es be Se Se Se Rohe Me Cote te te hot te a fo toto to roe tk Per: 


ro | 


a 


a pe 


AND EXTERNAL SECANTS. 


~ 


Ex. sec. 


SD -yo ok wrmwoRic | 


59 


10 | 


Vers. | D.1. 

9 724709 | 50 | 10.058099 
~ 724919 | 50 | .053547 
. 725129 50 053995 
"725339 | 8.50 |  .054443 
"725549 | 3.50 |  .054892 


725759 
. .725969 
726179 


. 726388 


va 
2-2 
5.29 
5 
U — 
co) 

Q 


9.731193 | 


731401 
. 731609 
731817 
. 732025 
. 182233 
. 732441 
732648 
732356 
‘ 


. 733064 


| 9.733271 


T8047 
. 738686 
783893 
. 734100 
.7384307 
784515 
.1384721 
. 73492 
785185 
9.735342 
730549 
735755 
735962 
.736169 
736875 
.736581 


+ 


7386788 
736994 


> : 
GO | 9.737200 


He ee ee 
JOUN AEN 


mH ee 
Ot Olas 


me eo 
coor O1rordr 


Cor 


PP Pop wp 
O39 OVOD C9 Oe 


SepeeseeweneneweD co 09 0D CN CH C9 CO CN 09 09 co Co 09 OD 9 OD CO 
a 
2 


iS) 
me me 
Co oe 


.0553840 
.055788 
.056237 
.056685 


.057134 


057583 | 


10 .058032 
058481 


.058930 
-059379 
.059828 
.060278 
.060727 
061177 
.061626 
.062076 


; 10:062526 


.065678 
.066129 


.066580 | 


10.067030 
067482 
.067933 
068384 
.068835 


069287 


.069738 
.070190 
070642 
.071093 
10 .071545 
.071998 
.072450 
.072902 
.073354 
073807 
.074260 
074712 
.075165 
.075618 
10.076671 
076524 
076977 
0774381 
LOTT 88 4 
.078338 
.078792 
079245 
079699 
10.080153 


my AP Ry ATAT AT AE Aa ag ss SS NNN NNN HSIN SN MINNA NNN NNNNNRANNN 


Or or or Or cr 


(oe) 


Jets 


47 


and 


Orr d 


ren orargr Ororg 


IA OTH -2 OU OT OT 


= ~ 
CSODMDRAUP WMHS 


Vers. 


Ex. sec, 


9.737200 
737406 
737612 
737818 
. 738024 
. 138230 
738436 
. 738642 
738847 
739053 


139258 
9.739464 


. 739669 
39875 
40080 
740285 
740490 
40695 
740900 
. 741105 
741310 
9.741515 
~TALT19 
TAL924 
 T42129 
142333 
(42588 
142742 
742946 
.743150 
743355 
9.743559 
(48762 
. 743967 
44171 
744875 
T4457 
744782 
744986 
745189 
745893 
9.745596 
. 745800 
746003 
746206 
. 746409 
7466138 
146816 
747019 


AIR | 


TATARA 
9.747627 
(47830 
(48033 
48235 
“748438 
748640 
. 748843 
749045 
749247 


9.749449 


oo 


h ARR ARRR RAD Bw 


co) 


wewewwwowwi cwcocwmccwci;to¢ wtwocwietwcto 


He 


BPR RP a Be 


IOSOWOCNWS WKHWWWWNWKWWWNH WW 


Oe 


— 
oO 


10080153 
080608 
.081062 


082425 
082880 
083835 
083790 
084245 
084700 


10.085155 


086066 
.086522 
086977 
087433 
.087889 
.088845 
088801 


10. 089714 
090171 
080627 
.091084 
.091541 
.091998 
.092455 
. 092912 
.098387' 
.098827 

| 10.094285 

.094743 

. 095200 

.095658 

.096116 

.096575 


097491 
097950 
.098408 


10.098867 
099326 
099785 
100244 
100704 
101163 
101623 
102082 
102542 
103002 

10. 103462 
103922 
104382 


105803 
. 105764 
106224 
. 106685 
.107146 


10.107607 


| 081516 | 
081971 


.085611 


089258 


097033. | 


104843 


oe LES rn ents Sie et 8 DSI NNSA aistst Bg ya a aa aa te a es ietet SIV sIV VWs INI 


“ 


Vers. 


9.749449 
749652 
- 749854 
. 750056 
- 750258 
. 750459 
- 750661 
. 750863 
751065 
751266 
751468 

11 | 9.751669 

12 | .751871 

13 152072 

14 T52278 

15 T2475 

16 152676 

17 | 752877 

18 753078 

19 753279 

20 | .7538480 


21 | 9.753681 
22) .753881 

23 | .754082 
Ha 24 | 754283 

25 | .754483 
26 | .754684 
27 | .754884 
28 |} .755085 


ed 
SOHIRUR IHS | ~ 


29 155285 
30 . 155485 
; 31 | 9.755685 


32 | .755886 
33 | .756086 
Baie 34 | .756286 

ie 35 | .'756486 
Wi 36 | .756685 
wT 37 | .756885 
ah 38 |  . 757085 
39 | .757285 
40 | . 757484 
41 | 9.757684 
42 | . 757883 
43 | .758083 
44 | .758282 
45 | . 758481 
46 | .758681 
47 | .'758880 
48 | .759079 
49 | .759278 


50 59477 
51 | 9.759676 
52 159875 
53 160073 
5 760272 


55 | .760471 
56 | .760669 
57 | .760868 
58 | .761066 
59 | .761265 
60 | 9.761463 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


64° 65° 
D. 1’.| Ex. see. | D 1’. z, Vers. | D.1".| Ex. see. 
3.38 | 10.107607 Tel 0 | 9.761463 |.3.30 | 10.135515 
3.07 . 108069 7.68 ] .761661 | 3.32 .135984 - 
3/37 . 108530 “we 2 . 761860 | 3.30 . 186454 
3.0% .108992 7.68 3 . 762058 | 3.30 . 1869238 
3.35 . 109453 sf 4 . 762256 | 3.30 . 137393 
Ook .109915 | 7.7 5 762454 | 3.30 .187863 | 
Sy aye 180377 | 785 6 . 762652 | 3.30 . 138333 
3.00 .110889 | 7.7 if . 762850 | 3.28 . 1388803 
3.35 111301 eek 8 . 763047 | 3.30 . 139273 
3.00 .1117638 Visite 9 - 763245. | 3.30 .1389744 
3.35 . 112226 ne 10 . 763443 |-3.30 . 140214 
3.37 | 10.112688 | 7.7 11 | 9.163641 | 3.28 | 10.140685 
3.35 .113151 “ote 12 .768838 | 3.30 .141156 
3.35 .113614 Late 13 . 7640386 | 3.28 . 141627 
Bot .114077 | 7.72 || 14 . 764233 | 3.28 . 142098 
3.35 .114540 112 15 . 764430 | 3.30 . 142569 
3.59 .115003 G92 16 . 764628 | 3.28 . 143041 
3.35 . 115466 4.42 17 . 764825 | 3.28 . 143512 
3.35 .115929 Tate 18 . 765022 | 3.28 . 148984 - 
3.35 . 116393 Ub Me 19 . 765219 | 3.28 . 144456 
3.35 . 116857 Cite 2 . 165416 | 3.28 . 144928 
8.33 | 10.117321 Vite 21 | 9.765613 | 8.28 | 10.145400 
3.35 .117785 see 2% . 765810 | 3.28 . 145872 
3.35 .118249 Tesi 23 . 766007 | 3.28 . 146845 
Shas! lIS713: | 7 738 || Ba 766204 | 3.2 . 146818 
3.35 119177 var, 25 . 766401 | 3.27 .147290 
aya: .119642 | 72% 26 766597 | 3.28 .147763 
3.35 . 120106 at 27 . 766794 | 3.28 . 148236 
Side 120571 7 28 .766991 | 3.27 . 148710 
3.33 . 121036 CM 29 . 767187 | 3.28 . 149188 
3.33 .121501 v@ares) 30 . 767384 | 3.27 . 149657 
8.85 | 10.121966 | 7.75 || 831 | 9.767580 | 3.27 | 10.150130 
3.33 .1£2431 WAG || 82 S76(776- | 8227 .150604 
3.33 122897 | 795 || 338 167972. | 3.28 .151078 
3.33 . 123362 CHE 34 .768169. | 3.27 .151552 
3.02 . 123828 Ceres 35 . 768365. | 3.27 . 152027 
3.33 . 124294 TOU ve 36 . (68561. | 3.27 . 152501 
3.33 . 124760 CAG 37 68757 | 3.27 . 152976 
3.53 . 125226 7.70 || 38 . 768953. | 3.27 . 153450 
3.02 . 125692 Wh 39 . 769149 | 3.25 . 158925 
3.33 .126158 7.78 40 . 169344 | 3.27 . 154400 
3.82 | 10.126625 |. 7.78 || 41 | 9.769540 | 8.27 | 10.154876 
3.33 . 127092 Vsti |) 42 . 769736 | 3.25 . 1553851 
3.382 .127558 7.78 || 43 . 769931. | 3.27 . 155826 
8.382 . 128025 7.98 44 Stiletto. et . 156302 
3.00 . 128492 | 7.80 || 45 .770323 | 3.25 .156778 
3.32 . 128960 7.78 46 ~770518 | 3.25 . 157254 
3Jo8 . 129427 @fiom Wie Ve MOM8 || 8.2% . 157730 
3.32 . 129894 7.80 48 .770909. | 38.25 . 158206 
3.02 . 180862 7.80 49 .771104 | 3.25 . 158683 
3.32 . 1308380 7.80 50 ~ 701299 | 3.25 .159159 
3.32 | 10.131298 | 7.80 |} 51 | 9.771494 | 3.25 | 10.159636 
Susy .131766 7.80 52 .771689 | 3.25 .160113 
3.32 132234 7.80 53 . 771884 | 3.25 . 160590 
$.32 . 182702 7.80 54 -772079. | 8.25 . 161067 
3.30 . 133170 7.82 55 CR2274. | 8.25 .161545 
3.32 . 138639 7.82 56 . 772469 | 3.25 . 162022 
3.30 . 134108 / ey 57 . 772664 | 3.23 . 162500 
3.32 . 184577 7.82 58 772858 | 3.25 . 162978 
3.30 . 185046 ‘eises 59 . 7730538 | 3.25 . 163456 
3.80 | 10.1385515 7:82 (| 60 | 9.778248 | 3.23 | 10.163934 


AND EXTERNAL SECANTS. 


/ Vers Dp, 1° \dixewsec Dy I’ J Vers. |D. 1’ 

0 | 9.773248 | 3.23.| 10.168934 | 7.98 0 | 9.784809 | 3.18 

1 V73442 | 3.23 .164413 |. 7.97 1 .785000 | 3.18 

2 773686 | 3.25 .164891 7.98 2 .785191 | 3.17 

3 773831 |. 3:23 .165370 | 7.98 3 .7853881 | 3.18 

4 .774025 | 3.23 .165849 | 7.98 4 7185572 | 3.18 

5 . 774219 | 3.25 .166828 | 7.98 5 .(85763 | 3.17 

6 .174414 | 3.23 .166807 | 7.98 6 785953 | 8.18 

7 .774608 | 3.23 . 167286 8.00 fi .786144 |- 3.17 

8 .774802 | 3.23 . 167766 7.98 8 .786334 | 3.17 

:.9 .774996 | 3.23 . 168245 8.00 9 . 786524 | 3.18 

10 .775190 | 3.23 .168725 | 8.00 || 10 ~786715 | 3.17 

| 11 | 9.775384 | 3.22 | 10.169205 | 8.00 || 11 | 9.786905 | 3.17 
| 2 TUT? | 3.23 -169685 8.00 || 12 .787095 | 3.17 
13 i dae 8 ips) oa Re .170165 | 8.02 || 18 ~ 787285 | 3.17 
| 14 V75965 | 3.2% .170646 8.02 || 14 (87475 | 38.17 
15 776159 | 3.22 ih abe 8.00 || 15 .787665 | 3.17 
: 16 7763852 | 3.23 .171607 8.02 || 16 . 787855 | 38.17 
17 776546 | 3.22 £172088 | 8.02 || 17 .788045 | 3.17 

18 776739 | 3.23 .172569 | 8.03 || 18 788235 | 3.17 

19 776933 | 8.22 .173051 8.02 |} 19 .788425 | 3.15 

20 UT77126 | 3.22 .1735382 | 8.03 || 20 788614} 3.17 

21 | 9.777319 | 3.22 | 10.1%4014 | 8.03 || 21 | 9.788804 | 3.15 

29 777512 | 3:22 .174496 | 8.03 || 22 .788993 | 3.17 

23 77105 -| 3.238 .174978 | 8.038 || 28 .789188 | 38.15 

24 .777899 | 3.22 .175460 | 8.03 || 24 .789372 | 3.17 

25} .778092 | 3.22 .175942 | 8.05 || 25 . 789562 | 8.15 
| 26 .778285 | 38.20 .176425 8.03 || 26 .789751 | 8.15 
27 TBAT 3.22 . 176907 8.05 || 27 .789940 | 3.17 

| 28 778670 | 3.22 .177390 | 8.05 || 28 .790180 | 3.15 
29 778863 | 3.22 177873 8.05 || 29 .790319 | 3.15 

30 779056 | 3.20 .178356 8.05 || 30 .790508 | 8.15 

31 | 9.779248 |-3.22.| 10.178839 | 8.07 || 31 | 9.790697 | 3.15 

82 .779441 | 3.22 .179823 8.07 || 32 .790886 | 38.15 

33 .779634 | 3.2 179807 8.05 || 33 .791075 | 3.15 

34 |, .779826 | 3.20 .180290 | 8.07 || 34 .791264 | 8.15 

35 .780018 | 3.22 18077 8.08 ||} 35 791458 | 3.13 

36 .780211 | 3.20 . 181259 8.07 || 36 .791641 | 3.15 

3v .780403 | 38.20 .181743°} 8.07 || 37 .791830 | 3.15 

38 .780595 | 3.20 .182227 | 8.08 || 38 .792019 | 3.13 

39 .780787 | 3.22 182712 8.08 || 39 792207 | 3.15 

40 .78098C | 3.20 .183197 | 8.08 || 40 . 792396 | 3.138 

41 | 9.781172 | 3.20 | 10.183682 | 8 08 || 41 | 9.792584 | 3.18 

42 .781364 | 8.20 .184167 | 8.10 2 192772 | 3.15 

43 .781556 | 3:18 .184653 | 8.08 || 48 .792961 | 3.13 

44 -781747 | 3.20 .1851388 | 8.10 || 44 .793149 | 3.13 

45 .781939 | 3.20 .185624'| 8.10 || 45 2938887 4.3.13 
46 .782131 | 3.20 .186110 | 8.10 || 46 ~793525 | 3.15 

47 782823 | 3.18 .186596 | 8.10 || 47 793714 | 3.138 

48 782514 | 3.20 .187082 | 8.10 || 48 .798902 | 3.13 

49 .782706 | 3.18 .187568 | 8.12 || 49 .794090 | 3.12 

50 . 782897 | 3.20 .188055 | 8.12 || 50 -794277 | 38.13 

51 | 9.783089 | 3.18 | 10.188542 | 8.12 || 51 | 9.794465 | 3.13 

52 .783280 | 3.18 . 189029 8.12 || 52 .794653 | 3.13 

53 ~783471 | 3.20 . 189516 8.12 || 53 .794841 | 3.12 

54 .783663 | 3.18 .190003 | 8.13 || 54 .795028 | 3.13 

| 55 .783854 | 3.18 .190491 8.12 Bd .795216 | 3.13 

56 .784045 | 3.18 .190978 8.13 || 56 .795404 | 3.12 

5Y .784236 | 3.18 .191466 | 8.13 || 57 .795591 | 3.13 

58 784427 | 3.18 .191954 |} 8.15 || 58 795779 | 3.12 

| 59 784618 | 3.18 .192448 | 8.13 || 59 .795966 | 3.12 

| 60 | 9.784809 | 8.18 | 10.192931 8.15 || 60 | 9.796158 | 3.13 


Ex. sec, 


192931 


193420 
193908 
. 194397 
194886 
. 195376 
195865 
196355 
196845 
197335 
197825 


10. 


198315 


198806 
199297 
199788 
20027 

20077 

201262 
.201753 
202245 
202787 


10. 


203229 


203722 
204215 
204707 
- 205200 
. 205694 
. 206187 
. 206681 
207174 
. 207668 


10. 


208162 


208657 
.209151 
.209646 
210141 
210636 
2111381 
.211627 
212123 
212618 


10. 


213115 


.213611 
214107 
214604 
.215101 
215598 
.216095 
216598 
217090 
217588 


10. 


218086 


218585 
219083 
219582 
220081 


.220580 | 


221079 
22157 


10. 


222078 
222578 


Sqeqeqeg=gegeg= aaa. 


AF ad a 8 HE OT 2 OL OT CO Ot 


68° 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


69° 


D: 1s 


/ Mersin. pbs 1 )' Mxeseeas Dal / Vers. Ex. sec. |D.1” 
0 | 9.796153 | 3.18 | 10.222578 | 8.33 0 | 9.807286 | 3.07 | 10.252957 | 8.55 
| .796341 | 3.12 .220078 8.33 i .807470 | 38.07 .2538470 -| 8.55 
21° 796528 | 3.12 . 223578 8.3b° 1) 32 .807654 | 3.05 .2539838 | 8.57 
S5) o  796ta)| 3:12 . 224079 8.33 | 3 .807837 | 3.07 254497 | 8.55 
4 .796902 | 3.12 224579 8.35 || 4 .808021 | 8.05 .255010 | 8.57 
5 ~797089 | 3.12 . 225080 8.35 || 5 .808204 | 3.07 .255524 | 8.58 
6 797276 | 3.12 . 225581 8.3%. IG .808388 | 3.05 .256039. | 8.57 
fi -797463' | 3.12 . 226083 Srao lt a7 .808571 | 3.07 .256553 | 8.58 
8 .797650 | 8.12 . 226584 8.37 8 .808755 | 3.05 .251068 | 8.57 
9 197837 | 3.10 . 227086 8.37 i 2 . 808988 | 3.05 .257582 | 8.60 
10 .798023 | 3.12 .227588 8.37 || 10 .809121 | 3.07 .258098 | 8.58 
11 | 9.798210 | 3.12 | 10.228090 8.37 || 11 | 9.809305 | 3.05 | 10.258613 | 8.60 
12 .798897 | 8.10 . 228592 8.38 || 12 .809488 | 3.05 .259129 | 8.58 
13 . 798583 | 8.12 «229095 8.38 || 18 .809671 | 3.05 .259644 | 8.60 
14 .798770. | 3.10 . 229598 8.38 || 14 .809854 | 3.05 .260160 | 8.62 
15 |. .798956 | 3.10 . 230101 8.38 |} 15 .8100387 | 3.05, .260677 | 8.60 
16 .799142 | 3.12 .230604 8.38 || 16 810220 | 3.05 .261193 | 8.62 
17 .799829 | 3.10 .231107 8.40 || 17 .810403 | 3.038 261710 | 8.62 
18 .799515 | 3.10 .231611 8.40 || 18 .810585 | 3.05 - 262227 | 8.62 
19 .799701 | 3.10 .232115 8.40 | 19 .810768 | 3.05 .262744 | 8.63 
20 .799887 | 3.12 .232619 8.40 |) 2 .810951 | 3.05 .268262 | 8.62 
21 | 9.800074 | 8.10 | 10.233128 8 40 || 21 | 9.811134 | 3.03 | 10.263877 8.63 
22 .800260 | 3.10 . 283627 8.42 || 22 .811816 | 3.05 .264297 | 8.638 
93 .800446 | 3.08 . 2841382 8.42 23 .811499 | 3.03 .264815 | 8.65 
24 .800631 | 3.10 234637 8.42 24 .811681 | 3.05 .265384 | 8.65 
25 800817 | 3.10 .205142 8.42 || 25 .811864 | 3.03 .205858 | 8.63 
26 .801003 | 3.10 . 235647 8.48 || 26 .812046 | 3.03 .266371 | 8,67 
27 .801189 | 3.10 . 236153 8.42 || 27 .812228 | 3.038 .266891 | 8.65 
28 |. 801375 | 3.08 . 236658 8.43 28 .812410 | 3.05 267410 | 8.67 
29 .801560 | 3.10 . 237164 8.43 |; 29 .8125938 | 3.03 . 267930 | 8.65 
3 ,801746 | 3.08 2237670 8.45 || 30 812775 | 3.08 .268449 | 8.68 
31 | 9.801931 | 3.10 | 10.238177 8.43 || 31 | 9.812957 | 8.03 | 10.268970 | 8.67 
82 .802117 | 3.08 . 238683 8.45 82 .813139 | 8.03 .269490 | 8.68 
33 . 802302 | 3.08 .239190 8.45 33 .8138321. | 3.03 .270011 | 8.67" 
34 ,802487 | 3.10 . 239697 8.45 34 .813508 | 3.03 .270531 | 8.68 
35 . 802673 | 3.08 . 240204 8.47 35 .813685 | 3.02 271052 | 8.7% 
36 .802858 | 3.08 240712 8.45 || 86 . 813866 | 3.03 271574 | 8.68 
37 .803043 | 3.08 .241219 8.47 || 387 .814048 | 3.03 2262095 | 8.7 
38 .803228 | 3.08 6241727 8.47 || 38 .814230 | 8.02 212017 | Sag 
39 , 8034138 | 3.05 , 242235 8.48 || 39 .814411 | 3.03 .2738139 | 8.7% 
40 .803598 | 3.08 PART 8.47 || 40 .814593 | 3.08 .2738662 | 8.7 
41 | 9.808788 | 3.08 | 10.248252 8.48 41 | 9.81477 8.02 | 10.274184 | 8.72 
42 | .808968 | 3.08 . 243761 8.48 42 .814956 | 3.02 204707 | 8.72 
43 .804158 | 3.08 . 244270 8.48 || 48 .8151387 | 3.03 -2c0e200 | 8% 
44 . 804338 |. 3.07 244779 8.50 || 44 .815319 | 3.02 2051538 | 8.7 
45 .804522 | 3.08 . 245289 8.48 || 45 .815500 | 3.02 4O2Cb| 8: te 
46 .804707 | 8.08 245798 8.50 46 .815681 | 3.02 .276801 | 8.73 
47 .804892 | 3.07 . 246308 | 8.50.)| 47 .815862 | 3.08 atlaee | 8. re 
48 .805076 | 3.08 . 246818 8.52 48 .816044 | 3.02 207849 | 8.75 
49 .805261 | 3.07 2473829 8.50 || 49 .816225 | 3.02 . 2783874 | 8.75 
50; .805445 | 3.07 247889 8.52 || 50 .816406 | 3.02 248899 | 8.75 
51 | 9.805629 | 3.08 | 10.248850 | 8.52 || 51 | 9.816587 | 3.00 | 10.279424 | 8.75 
52 | .805814 | 3.07 . 248561 8.52 2 ,816767 | 8.02 . 279949 | 8.77 
53 | .805998 | 3.07 249872 | §.52 53 .816948 | 3.02 280475 | 8.75 
54 |) .806182 | 3.07 . 249883 8.53 || 54 .817129. | 3.02 .281000 | 8.78 
55 | .806366 | 3.07 2503895 8.53 || 55 .817310 3.00 281527 | 8.77 
56 | .806550 | 3.07 . 250907 8.53 | 56 .817490 | 3.02 . 282053 | 8.78 
57 .806734 | 3.07 251419 | . 8.55 || 57 .817671 | 3.02 .282580 | 8.77 
58 .806918 | 8.07 . 2519382 8.53 58 .§17852 | 3.00 .283106 | 8.80 
59 .807102 | 8.07 .202444 8.55 59 .818032 | 3.02 . 283634 | 8.78 
60 | 9.807286 | 8.07 | 10.252957 8.55 60 1 9.818278 | 3.00 | 10.284161 | 8.80 


AND EXTERNAL SECANTS 


CHIR OMH WWOHS | 


Vers. Delt iss sec. Ds 1" 4 
| 9.818213 | 3.00 | 10.284161 | 8.80 0 
.818393 | 3.00 284689 | 8.7 1 
818573 | 3.02 285216 | 8.82 2 
.818754 | 3.00 285745 | 8.80 3 
818934 | 3.00 286273 | 8.82 4 
819114 | 3.00 286802 | 8.82 5 
819294 | 3.00 .287331 | 8.82 || 6 
.819474 | 3.00 287860 | 8.82 ve 
819654 | 3.00 288389 | 8.83 8 
819834 | 3.00 288919 | 8.83 9 
820014 | 3.00 289449 | 8.83 || 10 
9.820194 | 3.00 | 10.289979 | 8.85 || 11 
820374 | 2.98 .290510 | 8.85 || 12 
820553 | 3.00 .291041 | 8.85 || 13 
820733 | 3.00 ,291572 | 8.85 || 14 
820913 | 2.98 .292103 | 8.87 || 15 
.821092 | 3.00 292635 | 8.85 || 16 
821272 | 2.98 .293166 | 8.87 || 17 
821451 | 3.00 293698 | 8.88 || 18 
821631 | 2.98 294231 | 8.88 || 19 
.821810 | 2.98 294764 | 8.87 || 20 
9.821989 | 2.98 | 10.295296 | 8.90 || 2 
822168 | 3.00 295830 | 8.88 || 22 
22348 | 2.98 | .296363 | 8.90 || 23 
822527 | 2.98 .296897 | 8.90 || 24 
822706 | 2.98 297431 | 8.90 || 25 
822885 | 2.98 297965 | 8.92 || 26 
823064 | 2.98 .298500 | 8.90 || 27 
823243 | 2.97 299034 | 8.93 || 2 
823421 | 2.98 299570 | 8.92 || 29 
823600 | 2.98 .300105 | 8.93 || 30 
9.823779 | 2.98 | 10.300641 | 8.92 || 31 
823958 | 2.97 301176 | 8.95 || 32 
824136 | 2.98 .301718 | 8.93 || 33 
824315 | 2.97 302249 | 8.95 || 3 
824493 | 2.98 302786 | 8.95 || 35 
824672 | 2.97 303323 | 8.95 || 36 
824850 | 2.97 303860 | 8.97 || 37 
825028 | 2.98 804398 | 8 97 || 3% 
825207 | 2.97 804936 | 8.97 || 39 
825385 | 2.97 .805474 | 8.97 | 40 
9.825563 | 2.97 | 10.306012 | 8.98 || 41 
825741 | 2.97 .306551 | 8.98 || 42 
825919 | 2.97 .807090 | 8.98 || 43 
826097 | 2.97 .307629 | 9.00 || 44 
826275 | 2.97 .308169 | 8.98 || 45 
826453 | 2.97 .308708 | 9.02 || 46 
826631 | 2.97 .809249 | 9.00 || 47 
,826809 | 2.97 | .3809789 | 9.02 || 48 
.826987 | 2.95 810330 9.02 || 49 
827164 | 2.97 810871 | 9.02 || 50 
9 827342 | 2.95 | 10.311412 | 9.02 || 51 
827519 | 2.97 811958 | 9.03 || 52 
827697 | 2.95 812495 | 9.03 || 53 
827874 | 2.97 313037 | 9.05 || 54 
828052 | 2.95 .813580 | 9.03 || 55 
828229 | 2.95 814122 | 9.05 |! 56 
828406 | 2.97 .814665 | 9.07 || 57 
828584 | 2.95 815209 | 9.05 || 58 
823761 | 2.95 .815752 | 9.07 || 59 
9.828938 | 2.95 | 10.316296 | 9.07 || 60 


Vers. 


9.828938 
829115 
829292 
829469 
829646 


829823. | 
.830000 | 


.830177 
830353 
.830530 
.830706 
.830883 
.831059 
. 831236 
.831412 
.831589 
.831765 
.831941 
.832117 
832293 
. 832469 
832645 
832821 
832997 
83317 
. 8383849 
8388525 
.833700 
.833876 
834051 
. 834227 


.834402 
.834578 
.834753 
. 834928 
. 835104 
. 835279 
.835454 
835629 
. 8385804 
.830979 
.836154 
. 836329 
.836504 
.836678 
. 836853 
.837028 
.8387202 
.837377 
.837551 
.837726 
.837900 
. 838075 


ie) 


© 


WD WWW WWW 


838249 | 


838423 | 


838597 
838771 
838945 
.839119 

839293 


9.839467 


2. 


WN WWNWNWWWND WNWWNWNNWWWwWwW WNWWWWWNWWWYW 


SO © 
Ww 


WNWWNWNWWNWNWW 
OOOO OO OS 


5 
cS) 


95 


SMOnwnwoww 


Ex. sec. 


10.316296 
.316844) 
817385 
.311929 
.318475 
.3819020 
.019565 
.820111 
.820658 
.3821204 
.O21751 

10.322298 
.3822845 
6820093 
.823941 
.324489 
.825038 
825587 
.826136 
.826686 
.82(235 

10.327786 
.828336 
828887 
.829438 
.829989 
.3830541 
.831093 
.381645 
.832198 
.882750 

10.333304 
.o308807 
.084411 
.034965 
.380520 
.336074 
.336629 
.387185 
.307¢41 
.8388297 

10 .3838853 
.3839410 
.839967 
.840524 
.841082 
.3841640 
.842198 
.842756 
.843315 
.843875 

10.344434 
844994 
345554 
.846115 
.3846676 
3347237 
.847798 
.848360 
.348922 

10.349485 


WoOooOooowwocse 


lol 
Jen) 


eomowm cs 


Cc 0 60 6 


Ss 
ie) 


9 


WOMMMOMMOMOO WHMOMmUwomwUownowowowo ovr 


ASAIN 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


~ 


Ex. sec, 


~ 


Vers. 


Ex. sec. 


¢ DrammwwHo | 


9.839467 


.839641 
.839815 
.839989 
.840162 


.840336 | 


.840510 
.840683 
.840857 
.841030 
.841204 


9.841377 


.841550 
841723 
.841896 
.842070 
, 842243 
. 842416 
.842589 
842762 
842934 


9.843107 


848280 
_ 843458 
843625 
84 8798 
848970 
.844143 
844315 
844488 
.844660 


9.844832 


845004 
845177 
845349 
845521 
845693 
845865 
.846037 
846208 
846380 


9.846552 


.846724 
. 846895 
.847067 
. 847238 
.847410 
.847581 
847753 
. 847924 
. 848095 
848267 
.848438 
. 848609 
848780 
.848951 
.849122 
.849293 
.849464 
. 849634 


9.849805 


IWWWNWWWWW 


09 0 


9 00 


DW 0 0 


G90 0000 2009 G DO OD< 
THT QO GS Tt 


OO 


OLOUd 


| 10.849485 


. 850048 
850611 
-do1175 
.851738 
802303 
. 852867 
303432 
.808997 
804563 
80012 


10. 355695 


.3896261 
.3856828 
.857395 


= -357963 


-008531 
. 809099 
.859668 
. 8602387 
. 3860806 
10361376 
.3861946 
.862516 
. 863087 
.863658 
.864229 
.864801 
.865373 
.805945 
.3866518 


10.367091 


. 867665 
. 3868239 
. 868813 
. 3869387 
.869962 
.8705388 
.871118 
.3871689 
.8(2266 


10.372842 


8738419 
.313997 
87457 
.875153 
.810731 
76310 
376890 


877469 


.8¢8049 


10.378630 


.879210 
.3 (9792 
880373 
. 3880955 
.881537 
382120 
882703 
. 883286 


10.383870 


OMNIAWIR WW © 


COOOowoeo 
TOV on en or etc = 


Vela ilelleelleilelisiielie) 


FITS ~ 
ww 


~ 


Ea 


RSE 


9.849805 
.849976 
.850147 
.850317 
. 850488 
. 850658 
. 850829 
.850999 
. 851169 
.851340 
.951510 


9.851680 
.851850 
.852020 
.852190 
.952360 
Sth 
- 852700 
.852870 
.853040 
. 8538209 


9.853379 
.853549 
.853718 
.853888 
. 854057 
. 854227 
. 854396 
. 854565 
. 854735 
. 854904 

9 855073 
.855242 
. 855411 
. 855580 
. 855749 
.855918 
856087 
. 856255 
856424 
. 856593 


| 9.856762 


. 856930 
. 857099 
.857267 
. 857436 
. 857604 
.857772 
.857941 
858109 
.858277 
9.858445 
. 858613 
. 858781 
.858949 
.859117 
859285 
~859453 
.859621 
859788 


80 
9.859956 | 2.80 


10383870 


884454 
3850388 
885623 
. 886209 
386794 
887380 
.387967 
. 888554 
.3889141 
389728 


10.390316 


-890905 
391493 
392082 
392672 
393262 
. 893852 
394443 
395034 
395625 


10.396217 


3896809 
.397402 
3897995 
898589 
.899182 
899777 
-400371 
. 400966 
.401562 


10.402158 


402754 
.403351 
. 403948 
404545 
.405143 
-405742 
. 406340 
406939 
407539 


10.408139 


.408739 
.409340 
-409941 
-410543 
411145 
411747 

-412350 
412954 
413557 


10.414161 


.414766 
415371 
.415976 
.416582 
417189 
417795 
.418402 
.419010 


10.419618 


a so Man Ai ae i iat ae 


CIRO mwwee | 


| 
| 
| 


1 


AND EXTERNAL SECANTS. 


ee 


Vers. |D. 1".| Ex. sec. | D.1".|| ’ | Vers. 1".| Ex. sec. |D. 1’ 
| 
9.859956 | 2.80 | 10.419618 | 10.13 | 0.) 9.869924 | 2.75 |10.456928 | 10.60 
"960124 | 2.78 | .420226 | 10.15 || 1] .870089 | 2.73 | .457564 | 10.62 
860291 | 2.80 "420885 | 10.17 || 2 | .870253 | 2.75 | .458201 | 10.638 
860459 | 2.78 "421445 | 10.15 || 3 | .870418 | 2.73 | .458839 | 10.62 
“860626 | 2.80 422054 | 10.17 || 4] .870582 | 2.75 | .459476 | 10.65 
860794 | 2.7% 422664 | 10.18 || 5 "0747 | 2.73 | .460115 | 10.65 
860961 | 2.7 423275 | 10.18 || 6 | .870911 | 2.75 | .460754 | 10.65 
861128 | 2.80 "493886 | 10.20 || 7% | .871076 | 2.73 | .461593 | 10.67 
,861296 | 2.78 "424498 | 10.20 || 8 | .871240 | 2.73 | .462033 | 10.67 
861463 | 2.78 -425110 | 10.20 || 9 | .871404 | 2.73 | .462673 | 10.68 
861630 | 2.78 425722 | 10.22 || 10 | .871568 | 2.73 | .463814 | 10 7 
| 9.861797 | 2.78 | 10.426385 | 10.22 || 11 | 9.871782 | 2.78 '10.463956 |. 10.70 
(861964 | 2.78 "426948 | 10.23 |) 12 | .871896 | 2.73 | .464598 | 10.70 
862131 | 2.78 "427562 | 10.28 || 13 | .872060 | 2.73 | .465240 | 10.72 
862298 | 2.78 428176 | 10.23 || 14 | .872224 | 2.73 465883 | 10.75 
(862465 | 2.78 "428790 | 10.27 || 15 | .872388 | 2.73 | .466527 | 10.73 
862632 | 2.78 "429406 | 10.25 || 16 | .872552 | 2.73 | .467171 | 10.73 
862799 | 2.77 "430021 | 10.27 || 17 | .872716 | 2.73 | .467815 | 10.75 
862965 | 2.78 "420637 | 10.27 || 18 | .872880 | 2.72 | .468460 | 10.77 
863132 | 2.78 431253 | 10.28 | 19 | .873043 | 2.73 | .469106 | 10.77 
863299 | 2.77 431870 | 10.3 | 20 | .873207 | 2.73 | .469752.| 10.77 
9.863465 | 2.78 | 10.432488 | 10.28 || 21 | 9.878871 | 2.72 |10.470398 | 10.78 
863632 | 2.78 | 433105 | 10.32 || 22 | .873534 | 2.7% 471045 | 10.80 
863799 | 2.77 433724 | 10.80 |: 93 878698 | 2.72 471693 | 10.80 
863965 | 2.77 "434342 | 10.32 || 24 | .873861 | 2.73 | .472341 | 10.82 
864131 | 2.78 "434961 | 10.33 || 25 | .874025 | 2.72 | .472990 | 10.82 
864298 | 2.77 "435581 | 10.33 || 26| .874188 | 2.72 | 473639] 10.88 
864464 | 2.77 436201 | 10.33 || 27 “4351 | 2.73 , .474289 | 10.83 
834630 | 2.78 436821 | 10.35 |; 28 4515 | 2.72 | .474939 | 10.85 
864797 | 2.77 437442 | 10.37 || 29 | .874678 | 2.72 | .475590 | 10.87 
864963 | 2.77 438064 | 10.37 || 80 | .874841 | 2.72 | _ 476242 | 10.85 
9.865129 | 2.77 | 10.438686 | 10.37 || 31 | 9.875004 | 2.72 |10.476893 | 10.88 
"965205 | 2.77 |  .439808 | 10.38 || 82 | .875167 | .2.72-| .477546 | 10.88 
"865461 | 2.77 | 489981 | 10.38 || 83 | .875320 | 2.72 78199 | 10.88 
865627 | 2.77 440554 | 10.40 || 34 | .875493 | 2.72 | .478852 | 10.90 
865793 | 2.77 "441178 | 10.40 || 35 | .875656 | 2.72 | .479506 | 10.92 
865959 | 2.75 "441802 | 10.42 || 36] .875819 | 2.72 | .480161 | 10.92 
866124 | 2.77 442427 | 10.42 || 87 | .875982 | 2.72 | .480816 | 10.93 
866290 | 2.77 "443052 | 10.43 || 88 | .876145 | 2.72 | .481472 | 10.93 
866456 | 2.77 "443678 | 10.43 || 39 | .876308 | 2.70 | .482128 | 10.95 
866622 | 2.75 "444304 | 10.45 || 40 | .876470 | 2.72 | .482785 | 10.95 
| 9.966787 | 2.77 | 10.444931 | 10.45 || 41 | 9.876633 | 2.72 |10.483442 | 10.97 
866953 | 2.75 445568 | 10.45 || 42 | .876796 | 2.70 | 484100 | 10.98 
867118 | 2.77 "446185 | 10.47 || 43 | .876958 | 2.72 | .484759 | 10.98 
867284 | 2.75 "446813 | 10.48 || 44] .877121 | 2.70 | .485418 | 10.98 
867449 | 2.75 447442 | 10.48 || 45 | .877283 | 2.7 486077 | 10.98 
867614 | 2.77 448071 | 10.48 |, 46 | 877445 | 2.72 | .486738 11.00 
867780 | 2.75 448700 | 10.50 || 47 | .877608 | 2.7 487398 | 11 02 
867945 | 2.75 "449330 | 10.52 || 48 | .877770 | 2.70 | .488059 | 11.03 
868110 | 2.75 449961 | 10.52 || 49 | 877982 | 2.7% 488721 | 11.05 
868275 | 2.77 450592 | 10.52 || 50 | .878095 | 2.7 489384 | 11.05 
| 9.969441 | 2.75 | 10.451228 | 10.53 } 51 | 9.878257 | 2.70 |10.490047 | 11.05 
"968606 | 2.75 L 451855 | 10.53 || 52} .878419 | 2.70 | .490710 | 11.07 
868771 | 2.75 452487 | 10.55 || 53 | .878581 | 2.7 491374 | 11.08 
868936 | 2.73 "453120 | 10.57 || 54 | .878743 | 2.7 492039 | 11.08 
869100 | 2.75 453754 | 10.57 || 55 | .878905 | 2.7 492704 | 11.10 
869265 | 2.75 "454383 | 10.57 || 56 | .879067 | 2.7 493370 | 11.10 
869430 | 2.75 "455/22 | 10.58 || 57 | .879229 | 2.68 | .494036 | 11.12 
869595 | 2.95 455657 | 10.58 | 58 | -.879390 | 2.70 | .494703 11.13 
869760 | 2.7 456292 | 10.60 || 59 | .879552 ) 2% 495371 | 11.13 
2 


9.869924 | 2.7 


10.456928 


| 9.879714 


4 
70 


110. 496039 


Vers. 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


76° 


D. 1".| Ex, sec. 


~ 


cay 
CO 


COHIOwUkK wore 


ie) 


=) 


N=) 


ay 

| 
ThFy 

hae 
Panty 
Wy 
eT | 
Hien | 
My 


=) 


| 9.879714 


879876 
.880037 


.880199 | 
. 880360 

.880522. | 
- 880683 | 
.880845 | 


.881006 


8381167 


881329 
881490 
881651 


.881812 | 


831973 
882134 
882295 
882456 


882617 


882777 
.882938 
.8383099 


883260 


883 120 
.8335381 
883741 
833902 


884062 | 
884223 


884383 
884543 
884703 
884864 
885024 
885184 
. 885344 
885504 
885664 
885824 
885983 
886148 
886303 
886463 
886622 
886782 
.886941 
-887101 
887260 
887420 
887579 
887739 


9.887898 
.888057 
.888216 
.888375 
888534 
.888693 
.888852 


.889011 


889170 


9.889329 


VWWWNWWw®W WWW WWWWWWWW 


Wwe 


9 0 WW WW WWW 


Se Se G2 G2 Gd Ao So 
on NW NN 


WWNWWWWNWWHDW WWWWKwwwwww we 
. Da O 
oe = 


POA 2A2R20RRR0 


WW WD 


10.496039 


68 496707 


4973877 


.68 |  .498047 
00 498717 
.68 .499388 
70 .500060 
.68 -900732 
.68 .501405 


502078 


68 502752 


.68 | 10.503426 
.68 .904102 
.68 504777 
63 505454 
68 .506131 
68 506808 
.68 507486 
67 .508165 
.63 503844 
8 009524 
68 | 10.510205 
67 .910386 
68 .911568 


67 512250 
68 912933 
67 .513617 


68 .914301 
67 .514936 


515572 
516358 
10.517045 
517732 
7 | .518420 
7 | 1519109 
7 | 1519798 
7} 520188 
521179 
521870 
7 | .522562 
523254 
10.523947 
524641 
7 | 525335 
5 | 526030 
7 | — 526726 
5 | 527423 
7 | 528120 
5 | 528817 
7 | .529516 
5 | 530215 
5 | 10.530914 
5 | 531614 
5 | 582315 
5 |  .583017 
65 |  .533719 
165 | 534422 
2.65 | 535126 
2.65 | 535830 
2.65 .586535 


2.65 | 10.537241 


AIRHOR WOH S 


59 


Vers. ol Exec, 
9.889329 | 2.65 10.537241 
.889488 i 537947 
889647 588654 
889805 .539362 
889964 540071 
89012: 540780 
890281 541490 
890440 .542200 
.890598 542911 
890757 543623 
890915 544336 
9.891073 '10.545049 
891232 545763 
.891390 546477 
891548 547193 
891706 547909 
891864 .548626 
892022 549343 
.892180 “550061 
.892338 .550780 
892496 .551500 
9.892654 | 552220 
892312 | 552941 
892969 .553663 
893127 554385 
893235 .555109 
893142 555833 
. 893600 556557 
893753 | 557283 
893915 .558009 
894072 | .558736 
9.894230 10.559463 
894387 560192 
894544 560921 
894702 561651 
894859 562381 
895016 563113 
895173 563845 
.895330 564577 
895487 5653811 
895644 566045 
9.895801 '10.566781 
.895958 567516 
896115 568253 
896272 .568990 
.896428 569729 
.896585 .570468 
896742 571207 
.896898 .571948 
897055 .572689 
897211 573431 
9.897368 10.5741'74 
897524 574917 
.897680 515662 
897837 576407 
897993 BV7153 
.898149 .5Y7900 
.898305 578647 
898461 579396 

9 | 898618 580145 
60 | 9.898774 110580895 


=o 
IWW WWW 


~ 


2S 


~ 


Wd 


BW 09 WWW 


eo 


Pah ek ah pak pk fk fal fe ek fk Pe Be pe 


ti) 


et et 
WW WNWNNNWNWWWe 
B09 COC wwe 


© 
~ 


fmt peek peek ped Pe ek pe 


9 WWW WW 


AND EXTERNAL SECANT*S. 


DORDIOUPFWWH OS 


10 


59 
60 


"| Ex. sec. 


Vers. 


| 9.898774 


. 898930 
. 899086 


.899709 


| .900176 
| 900334 
| 9.900487 
| ,900642 
| .900798 
| .900953 


. 91108 


| 901264 
| .901419 
| 90157 


. 901729 


.961884 
9.902040 


| 902350 
| 902504 
. 902659 
. 902814 
. 802969 
.908124 
.908278 
.903433 


.903742 
. 903897 
.904051 
. 904206 
~ 904360 
904514 
. 904668 
, 904823 
.90:977 
905131 
. 905285 
.905439 
.905593 


ile) 


5 | 905747 


.905901 
-906055 
.906209 
. 906363 
.906516 
9.906670 
. 906824 


907284 
.907438 


9.908051 


@ 0 %~ 


899241 | 
.899397 | 
899553 | 


9% 


.899865 
F 900020 


WWWWW We 


SWWWNWNWWNwWNww 


.902195 


KQQ | 
9.903588 


.906977 
907131 


WWWNWWNNWWNOW WWW WWNWNWNWYD WBUDWWWNWWNWNWW NWNWNWWWNWWOWWW 


Oorergrg 


.907591 
907744 | 
907898 


Or Oorsz Oud 


| 10.580895 


.581645 
582397 
.583149 
588903 
584657 


.585411 


.586167 


.586928 


.587681 
.588439 
10.589198 
.589957 
.590718 
.591479 
592242 
.598005 
.593769 
594533 
-595299 
.596066 


| 10.596833 


597601 
598370 
.599140 
599911 
“600682 
.601455 
602228 
603003 
603778 
10. 604554 
“60533 
.606108 
606887 


607667 | 


608447 
609228 
-610010 
610794 
611578 


10.612363 
613148 
.613935 
614723 
.615511 
.616301 
617091 
617883 
.618675 
.619468 


57 | 10.620262 


.621057 


.621853 


.622650 


.623448 
.624247 


625047 
625848 


. 626650 
10.627452 


| 9.908051 


DW IAD>rkwwore © 


| 9.909734 


| 9.911259 


fed ped ek Pek ed ped Ped Be pe RR Pt 
wecwwwowuwwwuU wwuwwioe 
p09 09.09 09 G9 IW DWN 


908204 
.908357 
.908511 
. 908664 
. 908817 
.908970 
. 909123 
. 909276 
. 909428 
. 909581 


. 909887 
. 910039 
.910192 
.910345 
.910497 
. 910650 
.910802 
.910955 
911107 


.911412 
911564 
911716 
.911868 
912020 
912172 | 
912324 
912476 
912628 
9.912780 
912982 
.918084 
913235 
.918387 
.913539 
.9138690 
. 913842 
.918993 
914145 


9.914296 
914448 
.914599 
-914750 | 
914902 
.915053 
915204 


.915855 
.915506 
.915657 
9.915808 
. 915959 
.916110 
.916261 
.916412 
.916562 
.916713 
.9168C4 
.917014 


WWWNWWWWWWW WWWww 


9.917165 


9 57 


10. 627452 
628256 
. 629060 
.629866 
.630673 
.681480 
. 6382289 
.633098 
. 6383909 
.634720 
. 635533 

10.686346 
.637161 
.6387976 
. 688792 | 
.639610 
.640428 
.641248 
. 642068 
. 642890 
.643713 


10.644536 
.§45361 
.646186 
.647013 
.647541 
648670 
.649499 
.6503830 
.651162 
651995 

10.652829 
658664 
.654501 
.655388 
.656176 
.657016 
. 657856 
.658698 
.659540 
.660884 

10.661229 
.662075 
. 662922 
.663770 
. 664619 
-665470 | 
.666321 
607174 
. 668028 
. 668883 

10.669739 
.670596 
.671454 
672314 
673174 
.674036 
.674899 
.675768 
.676628 

10.677495 


wwe wo co Go te 
r 


Heh ek peek feed Pet Red Ped Fd Pt 
209 05 CO 
WwOrOtror 


oo 


wwIwwWwKNWwWWwWww wWiwwwwwwwe 
Oo ~ 2 Qi 20 N 


— 


DODO D MII 
OA W SANS 


mel ek fee pk pk pk fee fk et Pd 


HS 09 09.09 9 
Sea amieke 
mS toto te 


80° 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


Som WINS | om 


rary 
(ani lele oh) 


Vers. 


9.917165 
.917316 
.917466 
917616 
917767 
917917 
918068 
918218 
918368 
-918518 
. 918668 


9.918818 
.918968 
.919118 
919268 
- 919418 
919568 
919718 
919868 
.920018 
-920167 


9.920317 
920466 
.920616 
920766 
920915 
- 921064 
921214 
921363 
.921512 
. 921662 


.921811 
. 921960 
. 922109 
. 922258 
. 922407 
. 922556 
. 922705 
. 922854 
- 923003 
- 923152 


9.923301 
- 923449 
923598 
923747 
923895 
- 924044 
-924192 
924341 
- 924489 
924637 

9.924786 
- 924934 
- 925082 
925231 
925379 
925527 
92567 
925823 
925971 

9.926119 


Jes) 


Dal" at Ex sec! 
2.52 | 10.677495 
2.50 678362 
2.50 679231 
2.52 .680101 
2.50 -680972 
2.52 681845 
2.50 .682718 
2.50 . 683593 
2.50 684469 
2.50 . 685346 
2.50 -686224 
2.50 | 10.687104 
2.50 687985 
2.50 688867 
2.50 689750 
2.50 690634. 
2.50 691520 
2.50 692407 
2.50 693295 
2.48 -694185 
2.50 -695075 
2.48 | 10.695967 
2.50 .696861 
2.50 697755 
2.48 - 698651 
2.48 .699548 
2.50 . 400446 
2.48 .701346 
2.48 . 002247 
2.50 . 703149 
2.48 «704052 
2.48 | 10.704957 
2.48 . 705863 
2.48 10677 

2.48 - 07680 
2.48 . 708590 
2.48 . 709501 
2.48 - 710414 
2.48 .711828 
2.48 . 7122438 
2.48 713160 
2.47 | 10.714078 
2.48 ~714998 
2.48 715919 
2.47 - 716841 
2.48 717764 
2.47 . 718689 
2.48 - 719616 
2.47 . 720543 
2.47 ~ 721472 
2.48 722408 
2.47 | 10.723335 
2.47 . 424268 
2.48 725203 
2 AW 726139 
2.47 127077 
2.47 . 728016 
2.47 . 728956 
2.47 . 729898 
2.47 . 780842 


2.47 | 10.7381786 


81° 
D. 1’. f Vers. | D.1".| Ex. see. | D. 1" 
14.45 0 | 9.926119 | 2.47 |10.731786 | 15.78 
14.48 1 926267 | 2.47 | .7827383 | 15.78 
14.50 2| .926415 | 2.45] .733680 | 15.83 
14.52 3 | .926562 | 2.47 | .7346380 | 15.83 
14.55 4 | .926710 | 2.47 | .735580 | 15.87 
14.55 5 | .926858 | 2.47 | .736582 | 15.90 
14.58 6 | .927006 | 2.45 | .9787486 | 15.92 
14.60 7 | (927153 | 2.47 | 1738441 | 15.95 
14.62 8 | .927801 | 2.45 | .739398 | 15.97 
14.63 9 | .927448 | 2.47 | .740356 | 16.00 
14.67 || 10 | .927596 | 2.45 | .741316 | 16.02 | 
14.68 || 11 | 9.927743 | 2.47 |10.'742277 | 16.03 
14.7 12 | .927891 | 2.45 | .743239 | 16.08 
14.72 || 18 | .928088 | 2.45 | .%44204 | 16.08 
14.73 || 14 | .928185 | 2.47 | 745169 | 16.13 
14.77 || 15 | .9283833 | 2.45 | 746137 | 16.13 
14.7 16 | .928480 | 2.45 | .747105 | 16.18 
14.80 || 17 | .928627 | 2.45 | .'748076 | 16.20 
14.83 || 18 | .928774 | 2.45 | .749048 | 16.22 
14.83 || 19 | .928921 | 2.45 | .750021 | 16.25 
14.87 || 20} .929068 | 2.45 | .%50996 | 16.28 
14.90 |} 21 | 9.929215 | 2.45 |10.751973 | 16.30 
14.90 || 22] .929862 | 2.45 | .752951 | 16.33 
14.93 || 23 | .929509 | 2.45 | .753931 | 16.35 
14.95 |} 24 | .929656 | 2.45 | .754912 | 16.38 
14.97 || 25.| .929803 | 2.45 |] 755895 | 16.42 
15.00 || 26 | .929950 | 2.45 | .756880 | 16.43 
15.02 || 27 | .930097 | 2.43 | .%57866 | 16.47 
15.03 || 28 | .930248 | 2.45 | 758854 | 16.50 
15.05 || 29 | .980890 | 2.45 | - .759844 | 16.52 
15.08 |} 80 | .980587 | 2.43 | .760885 | 16.53 
15.10 || 81 | 9.930688 | 2.45 |10.'761827 | 16.58 
15.13 || 82] .930830 | 2.48 | .'7628292 | 16.60 
15.15 || 33| .980976 | 2.45 | -.768818 | 16.62 
15.17 || 3 .9381123 | 2.438 | .%64815 | 16.67 
15.18 || 385 | .931269 | 2.45 | .765815 | 16:68 
15.22 || 86 | .981416 | 2.481 .766816 | 16.72 
15.23 || 87 | .981562 | 2.43°| .767819 | 16.7% 
15.25 || 88 | .931708 | 2.45 | .%68828 | 16.77 
15.28 ||. 389 | .931855 | 2.43 | .769829 | 16.80 
15.30 || 40 | .932001 | 2.48 | 770887 | 16.82 
15.33 || 41 | 9.982147 | 2.43 |10.771846 | 16.87 
15.85 || 42 | .982293 | 2.43} 772858 | 16.87 
15.37 || 43 | .982439 | 2.48! 773870 | 16.92 
15 88 || 44°} .9382585 | 2.43 | .774885 | 16.95 
15.42 || 45 | .9827381 | 2.48 | .775902 | 16.97 
15.45 || 46 | .982877 | 2.48] .'776920 | 17.00 
15.45 || 47 | .983023°| 2.48 |° .777940 | 17.02 
15.48 || 48 | .983169 | 2:48.| . .778961 | 17.07 
15.52 || 49 | .988315:| 2.42 | 779985 | 17.08 
15.53 || 50 | 933460 | 2.43 781010 | 17.12 
15.55 || 51 | 9.983606 | 2.43 |10.782087 | 17.13 
15.58 || 52 | .933752 | 2.42 | .788065 | 17.18 
15.60 || 58 | .933897 | 2.48] .784096 | 17.20 
15.63 || 54 | .934043 | 2.43] .785128 | 17.93 
15.65 || 55 | .934189°| 2.42 | .786162 | 17.97 
15.67 || 56 | .934834 | 2 48] .787198 | 17.30 
15.70 || 57 | .984480°| 2.42 | .788236 | 17.33 
15.73 || 58 | .984625°| 2.42 | .789976°| 17.35 
15.73 || 59 | .984770 | 2.438 | .7903817 | 17.40 
15.7 60 | 9.934916 | 2.42 110.791361 | 17.42 


oa 


82° 


~ 


Vers. 

9.934916 
. 935061 
. 935206 
. 935352 
.935497 
.935642 | 
935787 
. 9359382 
. 936077 
. 936222 
. 936367 
9.936512 
. 936657 
.936801 
.936946 
. 937091 
.937236 
.937380 
.937525 
. 937669 
.937814 
. 937958 
.938103 
. 9388247 
-938391 
2 - 938536 
26 . 938680 
27 | .938824 
28 . 938968 
29 .939112 
380 - 939257 


. 939401 
32 . 939545 
3: . 939688 
34 - 939832 
35 . 939976 
36 .940120 
37 . 940264 
38 . 940408 
39 . 940551 
40 .940695 


41 | 9.940839 

940982 
43 .941126 
44 941269 
45 .941413 
46 | .941556 
7 | .941699 


Ho | 


SO WH 2 C2 OU HP CO 0D 


rt 


oo 
hoe 
© 


> 
4) 


DWMWNWNW Wee eRe ee ee 
OP WO COON WOH 
eo) 


WW WWW WWNWNWW WWWNWNWNWNWNWNWD WWW NWNHNWNWWND WW 


48 | .941843 
49°) .941986 | 
50 | .942120 


B1 | 9.942272 | 
52} .942415 
53. | .942559 
54| .942702 
55 | 942845 
| 942988 | 
57 | .943131 
58 | .943273 
59 | 943416 | 
| 60 | 9.943559 | 


or 
(=r) 


D. 1".| Ex. sec. 


*y 
~) 


2 


F] 


29 2 %~ 


WWWNWWWE 


WBWNWWNWWNWWWW 


TWWWNWWWYW 


. 792406 
193453 


. 797660 
- 798716 
799774 
.800835 
.801897 
10.802961 
.804027 
805095 
.806165 
807237 
.808311 
.809387 
.810465 
811545 
.812627 


10.813711 
814797 
815385 
816975 
818067 
819161 
820257 
821356 
822456 
823559 

10.824664 
82577 
826879 
.827990 
829104 
830219 
831337 
832456 
“83357 
834708 

10835829 
836957 
838088 
839221 
84035? 
841494 
842634 
843776 
844921 
846068 

| 10.847217 

818368 


851836 
852997 
854161 
8553826 
856494 
10.85,7665 


| 10.791261 


194502 © 


. 849522 


.850678 | 


ee ee es 


FEA SI RPS SES SS ST ST St SS 2 


rae 
Seen eto} eS 


AND EXTERNAL SECANTS. 


Vers. 


9.943559 
943702 
. 943845 
943987 
. 944130 


944273 | 


944415 


944558 


944700 
944843 
944985 


9.945127 
945270 
945412 
945554 
945696 
945838 
945981 
946123 
946265 
- 946407 

9.946549 
946690 
946832 
946974 
947116 
947258 
947399 
940541 
947683 
947824 


9.947966 
.948107 
948249 
948390 
948531 
.948673 
.948814 
948955 
.949096 
949237 


9.949379 
949520 
949661 
949802 
. 949943 
. 950083 
. 950224 
. 9503865 
. 950506 
950647 

9.950787 


950928 


“951069 | 
951209 | 


951350 
. 951490 
951631 
951771 
. 951911 
9.952052 


Pils. 


0 WWM 
eo ao a0 
J 


2.37 


Co Ce G2 09 C9 C2 C2 
a AININNONN 


we) 


37 


WWNWWNWNWWWWW WWWWWWwWWwWwW WWNVNWNWNWNWWWD WNW WNNWWWWD WWNWwwnwnwnw 
5 Rimaroe 3 oo 
ea 


Ex. sec. | D. 1". 


)10.857665 | 19.55 
.858838 | 19.58 
.860013 | 19.68 
.861191 | 19.67 
862371 | 19.72 
.863554 | 19.7 

.864739 | 19.80 
.865927 | 19.83 
.867117 | 19.88 
.868310 | 19.92 
.869505 | 19.97 


10.870703 | 20.00 
.871903 | 20.05 
.873106 | 20.10 
.874312 | 20.18 
.875520 | 20.18 
876731 | 20.28 
-877945 | 20.27 
.879161 | 20.30 
.880379 {| 20.37 
.881601 | 20.40 


10882825 | 20.45 
"884052 | 20.48 
“885281 | 20.55 
-886514 | 20.58 
887749 | 20.62 
"888986 | 20.68 
890227 } 20.72 
891470 | 20.77 
"892716 | 20.82 
893965 | 20.87 


10.895217 | 20.92 
896472 | 20.95 
897729 | 21.00 
.898989 | 21.07 
. 900253 | 21.10 
.901519 | 21.15 
.902788 | 21.20 
.904060 | 21.25 
.905335 | 21.30 
.906613 | 21.33 


10.907893 | 21.40 
.909177 | 21.45 
.910464 | 21.50 
.911754 | 21.55 
.918047 | 21.60 
.914843 | 21.65 
915642 | 21.7 
.916944 | 21.75 
.918249 | 21.52 
.919558 | 21.85 


10.920869 | 21.92 
922184 | 21.97 
.923502 | 22.02 
924823 | 22.07 
926147 | 22.13 
927475 | 22.17 
.928805 | 22.23 
.9380139 | 22.30 
931477 | 22.33 


10.932817 22.40 


a eer a TS = oa —— 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


| | 
d : Vers. lO ass ella eal seas <5) 4 BA , Vers; | D. 1°.:! Ex. sees Dil" 
0 | 9.952052 | 2.33 | 10.932817 | 22.40 0 | 9.960397 | 2.30 |11.020101 | 26.40 
1 .952192 | 2.83 .984161 | 22.45 1 .960535 | 2.28 .021685 | 26.48 
2 952382 | 2.35 .935508 | 22.52 2 .960672 | 2.30 | .028274 | 26.57 
3 . 952473 | 2.33 -9386859 | 22.57 || 38 .960810 | 2.380 .024868 | 26.65 
4 .952613 | 2.33 .9388213 | 22.62 4 .960948 | 2.30 .026467 | 26.78 
5 .952753 | 2.83 .939570 | 22.68 5 .961086 | 2.28 .028071 | 26.80 
6 .952893 | 2.33 .940931 | 22.75 6 . 961223 | 2.30 .029679 | 26.90 
fi .9530383 | 2.83 .942296 | 22.78 || 7 .9613861 | 2.2 .031293 | 26.98 
8 .953173 | 2.33 .9438663 | 22.85 || 8 .961498 | 2.80 .032912 | 27.07 
9 .953313. | 2.33 .945034 | 22.92 9 .961636 | 2.28 .034536 | 27.13 
10 .9534538 | 2.33 . 946409 | 22.97 || 10 .961773 | 2.30 .036164 |. 27.28 
11 | 9.953593 | 2.32 | 10.947787 | 23.03 || 11 | 9.961911 | 2.28 |11.037798 | 27.38 
2 .9537382 | 2.33 .949169 | 23.08 || 12 .962048 | 2.30 .039438 | 27.40 
lies -953872 | 2.3: .950554 | 28.15 || 13 .962186 | 2.28 .041082 | 27.50 
ile’ .954012 | 2.33 .951943 | 28.22 || 14 . 9623823 | 2.2 .042732 | 27.58 
15 .954152 | 2.32 .953336 | 23.27 || 15 . 962460 | 2.28 .044387 | 27.67 
16 .954291 | 2.33 .954732 | 23.33 || 16 . 962597 | 2.80 .046047 | 27.77 
We .954431 | 2.33 .956132 | 23.38 || 17 .962735 | 2.28 .047713 | 27.85 
18 .954571 | 2.32 .957585 | 28.45 || 18 . 962872 | 2.28 .049384 | 27.93 
19 .954710 | 2.38 . 958942 | 23.52 || 19 .963009 | 2.2 .051060 | 28.08 
20 .954850 | 2.32 .960353 | 28.57 || 2 .663146 | 2.2 .052742 |. 28.13 
21 | 9.954989 | 2.33 | 10.961767 | 23.65 || 21 | 9.963283 | 2.28 |11.054430 | 28.22 
Q2 .955129 | 2.32 .963186 | 23.7 22 .963420 | 2.28 .056123 | 28.30 
93 .955268 | 2.32 .964608 | 23.77 93 . 963557 | 2.28 .057821 | 28.40 
24 .955407 | 2.33 .966034 | 23.82 || 24 . 963694 | 2.28 .059525 | 28.50 
25 955547 | 2.32 .967463 | 23.90 || 25 .968831 -| 2.2% .061235 | 28.60 
26 .955686 | 2.32 .968897 | 23.95 || 26 .963968 -| 2.27 .062951 | 28.68 
Q7 .955825 | 2.32 .970384 | 24.02 || 27 .964104 | 2.28 .064672- | 28.78 
28 . 955964 | 2.32 OTL 75> |-24.10-1)-2 .964241 | 2.28 .066399 | 28.88 
29 .956103 | 2.338 .9738221°| 24.15 29 .964378 -| 2.28 .068132 | 28.98 
30 .956243 | 2.32 .974670°| 24.22 || 3 .964515 | 2.27 .069871 |. 29.08 
31 | 9.956382 | 2.32 | 10.976123 | 24.28 || 31 | 9.964651 | 2.28 |11.071616 | 29.18 
ae .956521 | 2.32 .977580 | 24.85 || €2 .964788 | 2.27 .078367 |. 29.28 
Be .956660 | 2.82 .979041-| 24.42 || 33 .964924 | 2.28 .075124 | 29.38 
34 .956799 | 2.80 .980506 | 24.48 || 34 .965061 | 2.27 076887 | 29.48 
35 .956987 | 2.32 .981975° | 24.55 || 85,| -965197 | 2.2% .078656 | 29.58 
36 957076 | 2.32 .983448° | 24.63 || 36 2965834.| 2.27 .080431 | 29.68 
Bi .957215. | 2.32 .984926 | 24.68 || 37 .965460 | 2.2 .082212 | 29.80 
38 195 7ad4 1° 2.382 .986407 | 24.77 || 88 .965607-| 2.27 .084000 | 29.90 
39 .9574938 | 2.30 .987893 | 24.83 || 39 965743 | 2.27 .085794 | 80.00 
40 .957631 | 2.32 .989883-| 24.90 || 40 .965879 | 2.28 .087594 | 380.12 
41 | 9.957770 | 2.82 | 10.990877 | 24.97 || 41 | 9.966016 | 2.27 |11.089401 | 30.22 
42 .957909 | 2.30 .992375 | 25.03 2 .966152 | 2.27 .091214 | 30.82 
43 .958047. | 2.82 .9938877: | 25.12 || 43 . 966288 | 2.27 .093033 | 30.43 
44 .958186 | 2.30 .995384 | 25.18 44 .966424 | 2.27 .094859 | 30.55 
45 .958324 | 2.32 .996895- | 25.27 || 45 .996560 | 2.27 .096692 | 80.67 
46 .958463 | 2.80 .998411 | 25.33 || 46 .966696 | 2.27 .098532 | 80.77 
47 .958601 | 2.30 .999931 | 25.40 47 . 966882 | 2.27 .100378 | 80.87 
48 .958739 | 2.82 11.001455 | 25.48 || 48 .966968 | 2.27 .102230 | 31.00 
49 .958878 | 2.30 .002984: | 25.52 49 .967104 | 2.27 .104090 | 31.12 
50 .959016 | 2.30 .004517 | 25.63 |; 50 967240 | 2.27 .105957 | 31.22 
51 | 9.959154 | 2.30 | 11.006055 | 25.70 || 51 | 9.967376 | 2.27 |11.107830 | 31.35 
52 .959292 | 2.32 007597 | 25.7 52 .967512-| 2.25 .109711 | 31.45 
53 | .959431 | 2.30 | 009144 | 25.85 || 53 | .967647 | 2.27] .111598 | 31.58 
54 . 959569 | 2.30 .010695 | 25.93 54 967783 | 2.2 113498 | 31.68 
5D 959707 | 2.380 .012251 | 26.00 5D 967919 | 2.25 .115394 | 31.82 
56 .959845 | 2.3 .013811 | 26.10 56 . 968054 | 2.27 .117808 | 31.98 
57 .959983 | 2.30 .015377 | 26.17 || 57 .968190 | 2.27 .119219 | 32.07 
58 .960121 | 2.30 .016947 | 26.23 || 58 .968326 | 2.25 .1211438 | 32.18 
59 .960259 | 2.380 018521 | 26.33 || 59 | .968461 | 2.27 | - .128074 | 32.80 
60 | 9.960397 | 2.30 | 11.020101 | 26.40 || GO 9.968597 | 2.25 |11.125012 | 32.43 


. 


—s 
SODAS wwe S | 


86° 


AND EXTERNAL SECANTS, 


87° 


a 


Vers. Dp. 1°.; Ex, see, 


9.968597 | 2.25 
.968782 | 2.27 
.968868 | 2.25 
969003 | 2.25 
969138 | 2.2 
. 969274 2. 25 
.969409 | 2.25 
.969544 | 2.25 
.969679. | 2.25 
.969814 | 2.25 
. 969949 | 2.25 

9.970084 | 2.27 
. 970220 | 2.2 
. 970354 | 2.25 
.970489 | 2.25 
. 970624 :| 2.25 
970759 | 2.25 
. 970894 | 2.25 
.971029 | 2.25 
.971164 | 2.23 
971298 | 2.25 

9.971433 | 2.25 
.971568 | 2.23 
971702 | 2.25 
971837 | 2.23 
2 DELO TE 2.25 
.972106.| 2.23 
. 972240 | 2.23 
. 972874 | 2.25 
. 972509 | 2.2% 
. 972643 | 2.23 

9.972777 | 2.25 
972912 | 2.23 
-973046 | 2.23 
.973180 | 2.23 
.973314 | 2.23 
978448 | 2.2% 
973582 | 2.28 
.973716 | 2.23 
.973850 | 2.2% 
.973984 | 2.23 


9.974118 
- 974252 
. 974386 22 


SR OE CS) 
~) 
(ou) 


974519 | 2.23 
974653 | 2.23 
974787 | 2.22 
974920 | 2.23 
975054 | 2.23 
975188 | 2.22 
975321 | 2.23 
9.975455 | 2.22 
975588 | 2.23 
975722 | 2.22 
D75855. | 2,22 
975988 | 2.23 
976122 | 2.22 
976255 | 2.22 
.976388 | 2.22 
976521 | 2.22 
9.976654 | 2.23 


| 
1”.| Ex. sec. 


Dibts: - Vers. |D. | DOT 
11.125012 | 32.43 || 0 | 9.976654 | 2.23 |11.257854 | 42.52 
126958 | 32.55 1 | .976788 | 2.22 | .260405 | 42.73 
.128911 | 32.7 2 | .976921 | 2.22 | 262969 | 42.95 
130873 | 32.80 3 | .977054 | 2.22 | .265546 | 43.20 
132841 | 32.95 4 | .977187 | 2.22 | .268138 | 43.42 
.184818 | 33.07 5 | .977320 | 2.20 | .270743 , 43.67 
136802 | 33.22 || 6 | .977452 | 2.22 | .273363 | 43.88 
138795 | 33.33 7 | 977585 | 2.22] .275996 | 44.15 
.140795 | 33.47 8 | .977718 | 2.22 | .278645 | 44.38 
142803 | 33.62 9 | 977851 | 2.22 | .281308 | 44.68 
.144820 | 33.73 || 10 | .977984 | 2.20 | . .283986 | 44.88 
11.146844 | 33.88 || 11 | 9.978116~-| 2.22 |11.286679 | 45.13 
.148877 | 34.02 || 12 | .978249 | 2.2: 289387 | 45.38 
.150918 | 34.17 || 13 | .978382 | 2.20 | .292110 | 45.65 
.152968 | 84.80 |} 14} .978514 | 2.22 | .294849 | 45.92 
155026 | 34.43°|| 15 68617 | 2.20 | .297604 | 46.17 
157092 | 34.60 || 16 | .978779 | 2.22 | .8003874 | 46 45 
159168 | 34.73 || 17 | .978912 | 2.20 | -.808161 | 46.72 
.161252 | 34.87 || 18 | .979044 | 2.22 | .305964 | 47.00 
163344 | 35.03 || 19 | .979177.| 2.2 .808784 | 47.27 
165446 | 35.17 || 20 | .979309 | 2.22 | - .811620 | 47.55 
11.167556 | 35.33 || 21 | 9.979442 | 2.20 |11.314473 | 47.83 
.169676 | 35.48 |} 22 | .979574 | 2.20 | .817348 | 48.13 
.171805 | 35.63 || 23} .979706 |. 2.20 | .320231 | 48.43 
173943 | 35.78 || 2. 9798388 | 2.20 | 323137 | 48.72 
.176090 | 85.93 || 25 | .979970 | 2.22 | .826060 | 49.02 
.178246 | 86.10 || 26 | .980103 | 2.20 | .829001 | 49.33 
.180412 | 36.27 || 27 | .980235 | 2.20 | .331961 | 49.63 
182588 | 86.42 || 28 | .980867 | 2.20 | .334939.| 49.93 
.184773 | 36.58 || 29 | .980499 | 2.20 | .387985 | 50.27 
.186968 | 36.7 30 | .980631 | 2.20 | .340951 | 50.58 
11.189173 | 36.90 || 31 | 9.980763 | 2.20 |11.843986 | 50.92 
.191387 | 87.08 || 82} .980895 | 2.18 | .347041 | 51.23 
.193612 | 37.25 || 83 | .881026 | 2.20 | .850115 | 51.58 
.195847 | 37.42 || 84 | .981158 | 2.20 | 858210 | 51.92 
.198092 | 37.58 || 35 | .981290 | 2.20 | . 856325 | 52.25 
.200347 | 87.77 || 86 | .981422 | 2.20 | .359460 | 52.62 
-202613 | 87.93 || 87 | 981554 | 2.18} .862617 | 52.95 
204889 | 38.12 || 88 | .981685 | 2.20 | .865794 | 53.32 
207176 | 38.28 || 89 | .981817 | 2.2 868993 | 53.68 
-209473 | 38.47 || 40 .981949 | 2.18 | .872214 | 54.07 
11.211781 | 88.67 || 41 | 9.982080 | 2.20 |11.375458 | 54.42 
.214101 | 88.83 | 42 | .982212 | 2.18 | .378723 | 54.80 
.216431 | 39.03 || 43 | .982343 | 2.20 | .382011 | 55.20 
218773 | 39.20 || 44 | .982475.| 2.18 | .885828 | 55.58 
221125 :| 89.42 | 45 | .982606 | 2.18 | .888658 | 55.97 
.223490 | 39.58 || 46 | .982787 | 2.20 | .892016 | 56.38 
.225865 | 39.80 || 47 | .982869 | 2.18 | .895899 | 56.80 
.229253 | 39.98 || 48 | .983000 | 2.18 | .398807 | 57.20 
230652 | 40.18 || 49 | .983131 | 2.18 | .402239 | 57.62 
.233063 | 40.38 || 50 | .988262 | 2.20 |. .405696 | 58.07 
11.235486 | 40.58 || 51 | 9.983394 | 2.18 |11.409180 | 58.48 
237921 | 40.78 || 52 | .983825 | 2.18 | .412689 | 58.93 
240368 | 41.00 || 53 | .983656 | 2.18 | .416225 | 59.38 
242828 | 41.20 | 54 | .988787 | 2.18 | .419788 | 59.83 
245300 | 41.42 || 55 | .983918 | 2.18 | .423878 | 60.28 
247785 | 41.63 || 56 | .984049 | 2.18 | .426995 | 60.77 
. 250283 | 41.83 || 57 | .984180 | 2.18 | .480641 | 61.25 
252793 | 42.07 |} 58 | .984311 | 2.18 | .434316 | 61.73 | 
255317 | 42.2 | 59 | .984442 | 2.18 | .438020 | 62.22 
11. 257854 | 42.52 || 60 | 9.984573 | 2.17 |11.441753 | 62.7 


, 


CORIO WWH OS 


10 


Vers. 


( 9.984573 | 
. 984703 
.984834 | 
. 984965 
985096 
- 985226 
985357 
985487 
985618 
985748 
985879 


986009 
-986140 
986270 
986400 
986531 

986661 
986791 
986921 | 
987051 

987181 


987311 
987441 
987571 
987701 
987831 
987961 
988091 
988221 

“988350 
988480 | 


. 988610 
. 988739 
. 988869 
988998 
. 989128 
989257 
. 989387 
.989516 | 
. 989646 
9897715 
. 989904 
. 990034 
.990163 
. 990292 
.990421 
990550 
.990679 
990803 
:990937 
.991066 


.991195 
.991324 

991453 

. 991582 
.991710 
.991839 
.991968 
. 992096 
. 992225 


9.992354 


ILWWWNWNWND NYOWNWNWNWNWWNWWW WNW WWWWNWWWWYW 


WWW WWWNWWW WNWNWNWNHWWWW WWW 


re 


2) 


=e 


@O W Wes 


5 
~ 


BD WW WW 


—_— 


Ex. sec. 


| 11.441753 
445517 
449311 
-453137 
-456994 
-460883 
-464805 
-468761 
472751 
470775 
480834 
.484929 
.489061 
493230 
-497437 
.501683 
.505968 
.510293 
.514659 
-519066 
.523516 
.528010 
532548 
537131 
.541760 
546437 
.551161 
555935 
560759 
565634 
570561 


5735542 
58057 

585670 
.590819 
.596027 
601295 
606625 
.612018 
617475 
622998 


628589 
634250 
639982 
645788 
651668 
657626 
663663 
669781 
675984 
682272 

688649 
695117 
701679 

708338 
- 715097 
721958 
728925 
736002 
743192 
750498 


q+ 


115. 29% 
; 9086 


9215 
9345 
9474 
9603 
9732 
9862 
9991 


120 


0249 
0378 
0507 
0636 
0765 
0894 
1023 
1152 
1281 
1410 
1539 
1668 
1797 
1925 
2054 
2183 
2312 
2440 
2569 
2698 
2826 
2955 
3083 
3212 
3340 
3469 
3597 

726 
3854 
3983 
4111 
4239 
4368 
4496 
4624 
4752 
4881 


5009 
5137” 


5265 


| 5393 


| 
| 
| 
| 


5521 


5649 || 


5777 
5905 
6033 
6161 
6289 
6417 
6545 
6673 
6801 


15. 30*! 


TABLE 20 LOS ee SIGNS AND EXTERNAL 


OOIHD OUP OF TH © 


Je) 


°) 


=) 


© 


Vers. 


9.992354 
. 992482 
.992611 
992739 
. 992868 
. 992996 
.993124 
993253 
.993381 
.993509 
-993637 


.993765 
993894 
-994022 
994150 
994278 
994406 
994534 
. 994662 
. 994789 
994917 


995045 
995173 
995301 
995428 
995556 
995683 
995811 
995939 
996066 
996193 


.996321 
. 996448 
. 996576 
. 996703 
. 996830 
- 996957 
. 997085 
997212 
997839 
. 997466 
997593 
997720 
997847 
:997974 
.998101 
998228 


998355 


.998481 

998608 

. 998735 
998862 
998988 
. 999115 
. 999241 
.999368 
. 999494 
, 999621 
999747 
| .999874 
10.000000 


.| Ex. sec. 


TWWWWNWWWWW ; 


© . 


eh beh bed peek pre peek pee ek rm et ek peek beat rk re eek frm pr 
[WOW wWWwWwWwWwo WWW WwoWww 


WWWWNNWWNWWW 


WWWWNWNWWD VWWWNHNWNNWWWOYWD WNW wWwnwnnwnww 


% 0 


WW WW Ww wwww 


{1.750498 


157925 
~ (65477 
. 773158 
.780973 
(88926 
- 797022 
805268 
-813668 
822229 


.880956 


11.839858 
.848940 
858211 
86767 
877351 
887239 
897350 


907697 | 


- 918290 
929141 


. 940264 
.951672 
. 963381 
-975408 
11.987'769 
12.000485 
.013578 
.027069 
.040984 
.0553852 
12.070202 
.085569 
.101490 
.118008 
.135168 
153024 
.171634 
.191066 
.211896 
.232712 
12.255116 
20872 


803674 


1 


are 


.8380129 


.808285 
.888375 
420686 
.455575 
-493490 
535009 


12.580893 


632172 


.690291 
. 757364 


836672 


12..933708 

13.0587 
234991 
536148 

Inf. pos. 


0878 
1005 
1132 
1259 
1386 
1518 
1640 
1767 
1894 
2020 
2147 
2274 
2401 
2527 
2654 
2781 
2907 
3034 
3161 
8287 
3414 
3540 
3667 
3793 
8920 
4046 
4172 
4299 
4426 


'15,81* 


TADLE XXVII.—NATURAL SINES AND COSINES, 


~ 


0° 


~ 


OH CD VD 


2 


449 


“i 1° 9° 8° | 4° . 

Sine |Cosin | Sine. aan Sine Cosin| Sine |Cosin 
One. || .01745 30985 | (99939 | | 05234 99863 | 0 6976 | .99756| 60 
One. |} 01774! .9 | 99938 | “05263 .99861 || .07005 | .99754| 59 
One. |) .01803} . 99937 | 05292 | .99860} 07034 | .997 D2) 58 
7; One, || .01832} . 80 . 99936 | | .05321 | .99858 || .07063| .99750) 57 
| One. || .01862 |. }| .99935 || 05350 | 99857 || .07092| .99748 | 56 
5| One. || 01891] .99982 9|.99934)| 05379 | 99855 || .07121} .99746) 55 
75| One. |; .01920] .99982 || 99933 | 05408 | .99854|| .07150| 99744 54 
One. |) .01949) .99981 || | .99932/| 05437} .99852!| .07179| .99742) 53 
One. |/..01978] .99980 | .99931 || .05466 .99851 || 07208] 99740. 52 
One. || .02007, .99 | 2| .99930 || .05495 | .99849|| .07237| .99733 | 51 
| One. || .02036 | . 9! || . 99929 | | .05524 | .99847 || .07266| .99736) 50 
20! .99999 || .02065 | . |): é . 99927 || .05553 | .99846 || .07295| .99734! 49 
349 | .99999 || .02094! . nee 05582 | .99844|| .07324| .99731) 48 
78! 99999 | “02123 (99977 || 3 - 99925 || .05611 | .99842|| .07853] .99729| 47 
7| .99999 || .02152} .99977 || 03897 "99924 | .05640 | .99841 || .07382] 99727) 46 
51. 99999 || 02181}. ||. 5| .99923|| .05669 | .99839'| .07411| .99725| 45 
00465 | 99999 || .02211 99976 ||. 55 | .99922|| .05698 | .99838 || .07440] .99723| 44 
5! .99999 | 02240! .99975 || .03984| .99921 || .05727! 99836 || .07469| .99721| 43 
' 99999 |! 02269 | .¢ 7 99919 || .05756 | 99834 || .07498] .99719| 42 
. 99998 || .02298 | 96 2|.99918}| .05785 | .99833 | “O7527 .99716| 41 
$2) .99998 || .02327 99917 || .05814 | .99831 || .07556| .99714| 40 
|99998 || .02356 | . 9 {| - | 99916) .05844 99829 | .07585 . 99712) 39 
99998 || 02385) 99972 || 04129) .99915!| .05873) .99827 | .07614) .99710) 38 
j bese epi ced || . 99913) | .05902] .99826 | .0'7643 | 99708 | 37 
3 | .99998 || .02443] . ie .99912/| .05931 | .99824'| .07672| 99705) 36 
99997} | “p42 | 9 9 99911 || .05960).99822 | . ae .99703| 35 
699997 | 02501 | . ¢ | -99910|| 05989} .99821 | .07730|.99701 | 34 
7 85 | 99997 || 02530) . 99 | 99909 || .06018 | .99819 | . i309 .99699 | 33 
.99997 || 02560! .99967 || . .99907 || .06047| .99817 || .07788 | .99696| 32 
| .99996 || .02589) . 999 99906 | || 06076 .99815 || 07817} .99694| 31 
|. 99996 || .02618| . . 99905 | | .06105 |. 99813 || .07846| .99692| 30 
2.99996 | -02647 | . 99904 || .06134) .99812'| .07875] .99689)} 29 
. 99996 || .02676 | . 96 -99902 || .06163 | .99810 | .07904| .99687| 28 
-99995 || .02705 |! . | . 99901 || .06192| 99808 | .07933) .99685 | 27 
99995 || .02734 | .999 . 99900 || .06221 | .99806 || .07962| 99683] 26 
| .99995 || .02763}. || . 99898 | | .06250 | 99804 || .07991| 99680) 25 
| 99995 | .02792|..9 || 04596 | .99897|| .0627'9 | .99803 || .08020} .99678 | 24 
5} .99994 || .02821 | .99960 | 565 |.99896 || .06308|.99801 || .08049/ 99676) 23 
9; .99994}| .02850}. 594 | 99894) | .06337'| 99799 | .08078 | 99673 | 22 
. 99994} .0287' .99893 || .06366 | 99797 | .08107| 99671) 21 
. 99993 || .02908 | « 99892 | | .06395 | .99795 | .08136 | .99668) 20 

| | || | 
.99993}| .02938 | .99957 82} .99890|| .06424| .997'93 | .08165] .99666| 19 
222° .99993 || .02967 | .9 |.99889 |! .06453| 99792 | .08194 . 99664) 18 
251 | + 99992 || .02996 | . 99955 | |. 99888 |; .06482].99790 | .08223) .99661 | 17 
D' 99992 || .03025 | .99954 || . 99886 |; .06511) 99788 | .08252| .99659 16 
9.99991 || .03054 | .99953 || 8} 99885 || .06540).99786 | .08281| 99657) 15 
. 99991 || .03083 | , 99952 || .04827 - 99883 ||. 06569 | 99784 | .08810|.99654| 14 
367 | .99991 || 03112] .99952 ||. 3} 99882 || .06598 | 99782 | .08339] 99652) 13 
3! 99990 || ,03141 | .99951 || .04885).99881 || 06627] 99780 | 08368] .99649) 12 
25 | .99990)| .03170} .999 |.99879'| .06656].99778 | .08397| .99647| 11 
. 99989 || 03199}. . 99878 | 06685 a 08426 | .99644| 10 
8399989 | .03228] .99¢ 2 | gog7e | 06714 99774 || .08455! 99642! 9 
99989 || .03257 | .99947 || 99875 | | 06743 90772 08484 | .99639|} 8 
542) .99988 || .03286 | 99946 || .99873 |) .06773| 99770 | .08513) 99637] 7 
.99988 || .03316} .99945 || .05059} .99872 | | "06802 . 99768 || .08542|.99635| 6 
.99987 || .033845 | . 99° . 99870 | | -06831 | . 99766 '| .08571 | .99632) 5 
99987 || .03374 | .99 : 99869) .06860..99764 | 08600] .99630) 4 
-99986 || .03403 | .99942 || .05 .99867 || .06889 99762 | .08629/.99627| 3 
37 | 99986 || .03432] .999- 75 | .99866 || .06918 .99760!| .08658 | 99625} 2 
716 .99985 || .03461 | .999¢ 5}.99864|| .06947 .99758 | .08687'|.99622} 1 
5! .99985 || .03490 99863 | .06976 .99756 | .08716!.99619| 0 
| Sine i| Cosin |. 8 (Cosin| Sine | Cosin | Sine || Cosin | Sine . 

a = 
89° || 88 87° see | 85° 


TABLE 
5° | 6° 7° 
/ = = — ae Ns > sates || 
Sine |Cosin |} Sine Cosin | Sine |Cosin | 
0 | .08716|.99619 |) .10453) .99452 | .12187) .99255 || 
1 | .08745/ .99617)| . 10482! .99449 || .12216) .99251 
2 08774! .99614|| .10511| .99446 | . 12245! .99248 
3.08803 .99612|| 10540) .99443 |} 12274] .99244 | 
4 | 08831} .99609|! . 10569) .99440 || .12302' .99240 
5 | .08860] .99607 || .10597| .99437 || 12331] .99237 
6 | .08889] .99604|| 10626} .99434)| .12360 99233 
7 | 08918] 99602) | 10655) .99431 |} .12389 99230 
8 | 08947] .99599|| .10684| .99428 | .12418 .99226 
9 | .08976; .99596|| .10713 | .99424 || .12447 99222) 
10 | .09005 | .99594 || .10742 99421 12476 99219 | 
11 | .09034) .99591 || .10771| .99418 || .12504; .99215 | 
12 | .09063| .99588] | .10800) 99415 || .12533) .99211 | 
13 | .09092) .99586|| .10829 | .99412|| .12562| .99208 
14 | .09121| .99583}| .10858| .99409 || .12591 | .99204 | 
15 | .09150! .99580}| .10887 | .99406 || .12620 | .99200 
16 | .09179| .99578]| .10916 | .99402 || .12649| .99197 | 
17 | .09208) .99575 || .10945 | .99399 || .12678 | .99193 
18 | .09237) .99572|| .10973 | 99396 || .12706 | 99189 | 
19 | .09266) .99570) | .11002) .99393 || .12735 | .99186 
20 | .09295) .99567 || .11031 | .99390 || .12764] .99182 | 
21 | .09324} .99564|! .11060] .99386 || .12793).99178 
22 | 09353] 99562 |) .11089|.99383)| .12822|.99175' 
23 | .09382| .99559]| .11118) .99380'} .12851 | 99171 | 
24 | 09411} .99556]| .11147! 99377 | .12880) 99167 
25 | .09440] .99553|| .11176| .99374'| .12908] .99163 
96 | .09469| .99551}| . 11205! .99370 | .12937 | .99160 | 
27 | 09498] .99548}| .11234!.99367 || .12966 | .99156 | 
28 | .09527 | .99545|| .11263| .99364 | .12995| .99152 | 
29 | 09556! .99542|| .11291| .99360 | .13024} .99148 | 
30 | .09585 | 99540) | .11320) 99357 || .13053) .99144 | 
31 | .09614! .99537)| .11349] .99354'| .13081 | .99141 
32 | .09642| .995341| 11378) .99351'| .13110} .99137 
33 | .09671] .99531]|| .11407| .99347)| .13139| .99133 | 
34 | 09700] .99528)| .11436|.99344'| .13168) .99129 
35 | .09729| .99526]) .11465| 99341 || .138197]| .99125 | 
36 | .09758| .99523]| .11494| .99337 || .18226 | .99122 | 
37 | .09787|-.99520) | 11523] .99334 || .13254] .99118 | 
38 | .09816] .99517|| .11552) .99331 |] .13283) .99114 | 
39 | 09845! 995141! .11580)| .99327 || .13312| .99110| 
40 | .09874| .995114 11609) 99324, 13341 | 99106 | 
41 | .09903] .99508}| . 11638] .99320 | .13370 .99102 | 
42 | 09932} .99506]| .11667 | .99317 || .13399| .99098 | 
43 | .09961| .99503|| .11696| .99314/| .13427 | .99094 
44 | 09990] .99500}| .11725)| .99310 || .13456 | .99091 | 
45 | 10019} .99497|| .11754|°99307 || .13485 | .99087 
46 | .10048| 99494) |..11783|.99303)| .13514 | .99083 | 
7 | .10077| .99491|| .11812}.99300)| .13543 | .99079 | 
48 | .10106| .99488)| .11840| .99297 || .13572 | .99075 
49 | .10135] .99485]| .11869| .99293 || .13600 | .99071 
50 | .10164| .99482)| .11898 99290, 13629 | .99067 
51 | .10192|.99479|| .11927 | 99286)! . 13658 | .99063 
52 | 10221 |.99476|| 11956 | 99283)! 13687 | .99059 
53 | 10250) 99473} | .11985|.99279 || .13716 | .99055 | 
54 | .10279| 99470] | .12014).99276 || .13744|.99051 
55 | .10808).99467|| .12043] 99272 || .13773 | .99047 | 
56 | .10337|.994641) . 12071 | .99269 || .13802| .99043 | 
57 | 10366! .99461]) .12100' .99265 | 13831 | .99039 | 
58 | .10395) 99458] .12129!.99262'| .13860 | .99035 | 
59 | 1042499455] .12158!.99258 || .13889|.99031 
60 | .10453 | .99452| .12187| 99255 || .13917| 99027) 
: Cosin | Sine | Cosin| Sine || Cosin | Sine | 
| 84° 83° 82° 


XXVII.—NATURAL SINES AND COSINES. 


ge 


g° 


Cosin | 


glo || 


80° 


450 


Sine |Cosin |} Sine |Cosin 
.13917| .99027 | .15643 | 98769) 60 
13946) .99023 || 15672) 98764) 59 
.13975| 99019} .15701 98760 58 
.14004| .99015 | .15730).98755 | 57 
| .14033} .99011|! .15758;.98751 5 
.14061 | .99006 | .15787) 98746. 55 
.14090 | .99002 || .15816|.98741 54 
.14119| .98998 || 15845! .98737 53 
. 14148} .98994 || .15873 | .98732. 52 
.14177 | .98990 || .15902).98728' 51 
-14205 | .98986 recat meen 50 
.14234 | .98982 |) .15959|.98718' 49 
.14263 | .98978 || .15988|.98714) 48 
.14292) .98973 || .16017|.98709' 47 
.14320 | 98969 || .16046|.98704 46 
.14349 | .98965 || .16074| 98700! 45 
.14378 | .98961 |} .16103|.98695! 44 
.14407 | .98957 || .16132) .98690| 43 
.14436 | 98953}; .16160) .98686) 42 
14464} .98948|) . 16189) .98681) 41 
14493 | 98944 || .16218) .98676 40 
14522| .98940/| .16246| 98671, 39 
.14551 | .98936 || .16275 | .98667| 38 
.14580 | .98931 || .16304 | .98662) 37 
.14608 | 98927 || .16333).98657) 36 
“14637 | .98923|| .16361|.98652) 35 
.14666| .98919}) .16390) 98648) 34 
.14695 | .98914|| 16419 .98643! 33 
.14723) .98910]| .16447 | 98638! 32 
.14752| .98906 |) .16476 | .98633) 31 
14781 | 98902), . 16505 | .98629) 30 
.14810! .98897 || .16533|.98624) 29 
.14838| .98893 || . 16562 | 98619} 28 
.14867! .98889}| .16591|.98614! 27 
.14896 | .98884|' . 16620) .98609) 26 
.14925 | .98880|) .16648| .98604) 25 
.14954 | .98876 || .16677'| 98600) 24 
14982} .98871 || .16706 | .98595) 23 
.15011 | .98867|| .16734, .98590 | 22 
.15040 | 98863 || . 16763 | .98585) 21 
.15069 | .98858 |) . 16792) .98580) 2 
.15097' | .98854 |) .16820).98575| 19 
.15126 | .98849 || .16849 .98570; 18 
.15155 | .98845|) .16878 | .98565) 17 
15184 | .98841 |! .16906:.98561, 16 
.15212 | 98836) .16935 98556 | 15 
.15241 | .98832|| .16964' 98551) 14 
.15270 | 98827! .16992| .98546| 13 
15299 | .98823 || 17021) .98541) 12 
.15327 | .98818 || .17050) 98536! 11 
-15356  .98814)| .17078 98531 10 
15385 | .98809)) .17107|.98526, 9 
15414! .98805 || .17186|.98521! 8 
15442) 98800 || .17164) 98516) 7 
“15471 | 98796 |! 17193) .98511| 6 
15500; 98791 |}..17222: .98506| 5 
15529. 98787 || .17250}.98501| 4 
15557 |. 98782}; 17279 .98496' 3 
15586. .987'78|; .17308 .98491| 2 
17615 .98773|| .17336| .98486} 1 
15643 .98769| | .17365!.98481} 0 
Sine Cosin ; Sine 


A i ia ile ti i i i a ett ltl ee le art er Oe 


TABLE XXVII.—NATURAL SINES AND COSINES. 


/ 


el | 


10° {| 11° a 2° 18° Il 14° 
/ ragits ee | ge = , > oe si / 
Sine Cosin|| Sine ‘Cosin | Sine Cosin | Sine |Cosin || Sine |Cosin 
0 | 17865 . 98481 i “19, 81}. 98163 | 203 ‘91 .97815 | .22495| 97437 || .24192|.97030] 60 
1 |.17393 .98476!| .19109|.98157 | .20820 .97809 | .22523|.97430|| .24220| 97023) 59 
2 | 17422 .98471 || 191388) .98152 20848 | 97803 | 122552 | 97424 24245 | 97015 | 58 
8 | 17451 .98466)) .19167) .98146 | 20877 | 97797 || .22580| 97417 || .24277 | 97008) 57 
4 }:17479 .98461)| .19195}.98140 | .20905! 97791 || .22608) .97411 |) 24 305 | 97001 | 56 
5 | .17508 .98455 | .19224|.98135 | .20933) .97784 || .22637 | .97404 || .24383 | 96994 | 55 
6 | 17537 .98450 | 19252] 93129 | .20962| 97778 | .22665 | 07398 || .24362|.96987) 54 
7 | 17565 .98445|| .19281| 98124.) .20990| 97772 | .22693 | .97391 || .24390|.96980| 53 
8 | 17594 .98440)| 19309) 93118 | .21019| 97766!) 22722] .97384)| .24418] 96973) 52 
9 | 17623 .98435 |} .19333|.98112 | .21047|.97760 | .22750|.97378|| .24446|.96966! 51 
10 | .17651 .98430)| .19306|.938107,) .21076).97754 | .22778| 97871 || £24474) .96959 | 50 
11 | .17680{.98425)| .19395;.98101 || .21104 ee | -22807 | .97365 || .24503 96952 49 
12 | .17708|.98420)| .19423| 98096 || .21182|.97742 | .22835 | .97358 || .24531|.96945) 48 
13 | 17737 |.98414!| .19452).98C90 | .21161 ‘oir 22863 | 97351 || .24559 | 96937 | 47 
14 | .17766!.98409 | .19181).98084 || .21189] .977 22892 | 97345 || .24587 | .96930| 46 
15 | 17794 |.98404)| .19503; .938079 || .21218 1193 . 22020 | .97338 || .24615 | .96923| 45 
16 | .17823 | .98399| |: 19533 | .98073 | .21246).97717 | .22948 | 97331 |) .24644} 96916) 44 
17 | 17852) .98394|) .19565) 98067 || .21275) 97711 | 22977 | 97825 || .24672). 96909) 43 
18 | .17880 |.98389!); 19595) .98961 || .21303|.97705 | .23005 | .97318|| .24700 .96902) 42 
19 | .17909 | 98383 | | 119823! 98956 | .21331|.97698 | .23033 | .97311 || .24728|.96894| 41 
20 | 17937 |.98378 | -19552) 98050 || .21360) .97692 | 23002 | .97304|| .24756 | .96887) 40 
31 | .17966 |.98373]| .19580| 98044 | .21388} .97686 23090 | .97298 || 24784 |.96880| 39 
22 | 17995 |.98368/! .19709| .98039 || .21417| .97680 | .23118|.97291 || .24813) 96873) 38 
23 | . 18023 | .98362!! .19737| 93033 :) .21445|.97673 | .23146 | 97284) .24841) .96866 | 37 
24 | 18052 |.98357 || .19765| .98927'| .21474| .97667)| .23175 |. 972 78 | | 24869 | .96858 | 36 
25 | .18081/.98352) 19794) .98021 || .21502) .97661 || .23203 | .97271 |) .24897| .96851 | 35 
QE | .18109 | .98347)| .19823] 938916) ) .21530) 97655 || "23231 | 97264 | 24925 | .96844! 34 
27 | 18138) .98341 || .19851|.98010 | .21559|.97648!| .23260|.97257)| .24954| 96837) 33 
28 | .18166|.98336!| .19880|.98004 | .21587| 97642 || .23238|.97251 ||, .24982| .96829) 32 
29 | .18195 | .93331)| .19908} .97998 | 21616} .97635 | .23316 |. 97244 || .25010 | . 96822 | 31 
30 | .18224 | 98325) 19937) 97992 || .21644| 97630 || .23345 | .972387 || .25088 | .96815 | 30 
81 | .18252|.98320)| 19965) .97987 || .21672 97623 || .23373| .97230|| .25066| .96807| 29 
32 | .18281|.98315 | .19994|.97981!| .21701| .97617 || .23401 | 97223) | .25094) .96800) 28 
33 | .18309|.98310) 20022) .97975 | .21729) 97611 )| .23429) 97217 || .25122) .96793 | 27 
34 | .18338/.98304) .20051|.97969') .21758) 97604 || .23458 | .97210 | (25151 |. 96786 | 26 
35 | .18367 | .98299'| .20079).97963 || .21786| 97598 | .23486 | .97203)) .25179) 96778) 25 
36 | 18395 | 93294)! .20108| 97958 | .21814| .97592'| .23514|.97196 || .25207| .96771| 24 
37 | 18424! 98288 || .20136) .97952 | .21843] .97585)| .23542} 97189 || .25235 | 96764! 23 
88 |..18452 98283!) .20165).97946 || .21871) 97579 || .28571|.97182 || .25263 | 96756) 22 
39 | .18481 | .98277 |! .20193! 97940; | .21899! 97573); .23599|.97176 || .25291| .96749| 21 
40 | .18509| 98272 | .20222).97934 | . 21928) .97565 || .23627 97169 | . 25820 | .96742 | 20 
41 | .18538 |;98267 |! .20250| .97928 || .21956) 97560 || .23656 |.97162'| .25348} .96734| 19 
2 | .18567 | .98261)) 20279) .97922 .21985 | 97553 || .23684 | .97155) | .25376) .96727| 18 
43 | 18595! .98256)) .20307| 97916 | 22013) 97547 || .238712).97148 || .25404| 96719) 17 
44 | 18624 | 93250) 20336] .97910 | .22041| 97541 || .23740| .97141|| .25432).96712| 16 
45 | .18652 | .98245!' .20364) 97905 | | 22070 | 97534 || .23769 | .97134|) .25460) 96705 | 15 
46 | .18681 | .98240;! 20393) .97390 || 22093 | 97528 || 23797 | 97127 || .25488 | .96697| 14 
47 | .18710| 98234); .20421 | .97893 )| 2212 26| 97521 || .23825|.97120)| .25516) .96690) 13 
48 | .18738) .98229 || .20450 97887 || .22155':.97515 || 23853 |.97113 || .25545| 96682] 12 
49 | .18767 | 98223); .20478| .97831 || .22183|).97508 || .23882 | .97106 | 25573) .96675 | 11. 
50 | .18795 | .98213)) . ined 91875 || .22212| 97502 -23910 | .97100 | .25601 .96667| 10 
5 | 19824| 93212 20535) .97369 | 22240 | .97496 || .23938 |.97093 || .25629:.96660; 9 
52 | .18852 98207); .20563 97863 | | 22268 |.97489 | .23966 | 97086 |) .25657- 96653) 5 
53 | 18881 | .98201 || .20592) .97857 || .22297 |,.97483 || .23995 | .97079 |; 25685 .96645) 7 
54 | 18910) .98196!! .20620) .97851 || 22325! .97476 || .24023] 97072 | .25713 .96638 6 
55 | .18988 |.98190 |) .20649| .97845 || .22353 |.97470 || .24051 | .97065 || .2 5741 ,.96630! 5 
56 | .18967 |.98185|| .20677) .97839 || .22382 |.97463 24079 | 97058 | |'.25769 296623 |. 4 
57 | 18995 .98179|| .20706! 97833 |) .22410°.97457 ||.24108 | .97051 ||. .25798 .96615) 3 
58 |.19024 | .98174 || .20734| .97827 |) 22488 .97450 ||. 24136 | 97044 || .25826 .96608 | 2 
59} .19052 .98168/| .20763) .97821 || .22467 | .97444 | |-.24164| .97037)| .25854 .96600; 1 
60 | 19081 .98163)) .20791 | 97815 || .22495 97437 oeeaeee fiend | .25882 .96593) 0 
| Cosin Sine | | Cosin | Sine | | (Cosin Sine || Cosin | Sine » | Cosin Sine : 


| 79° 


NOES ! 


| 


76° ll 


‘15° 


16 


ie) 


an 


18° 


TABLE XXVII.—NATURAL SINES AND COSINES. 


__19° 


Sine |Cosin 
27564 | .96126 | 
.96118 | 
96110! 
96102 | 
96094 | 
. 96086 | 
96078 | 
96070 || 
96062 | 
. 96054 | 
96046 | 


. 96037 
. 96029 
; 96021 
27955! . 
. 96005 
95997 
. 95989 
. 95981 
95972 | 
. 95964 
. 95956 
95948 
. 95940 
95931 
. 96923 
.95915 
28318) . 
95898 
.95890 
. 95882 


27592 | 
. 27620 
27648 | 
27676 | 
27704 | 
27781 
27759 
20787 | 
.27815' 
27843 | 
Ri 71! 
27899 | 
27927 | 


27983 | 
.28011 
28039 | 
.28067 
28095 | 
. 28123 | 
.28150 
28178! 
. 28206 | 
28234 | 
28262 | 
28290 | 


28346 | 
28374 | 
28402 | 


28429 | 
28457 | 
28485 | 
28513 | 
28541 
28569 | 
28597 
28625 
28652 | 
28680. 
28708 | 
28736 


28792 | 
28520 
28847 | 
28875 | 


28908 | 
28931 | 


. 28987 | 
29015: 
29042 

29070 

29098 | 
29126 | . 
29154 
29182 
. 29209: 


29287 | 


Cosin | 


95782 
28764 |. 
95766 
95757 
95749 
"951740 
95732 
95724 
28959. 
95707 
95698 
95690 
95681 
95673 | | 


95656 
. 95647 


. 95630 


96013 


95907 


95874 
. 95865 
95857 
. 95849 
. 95841 
. 95832 
. 95824 
. 95816 
. 95807 
95799 


95791 


9577 


95715 


95664 


95639 


Sine 


| 
| 
| 
| 


| .82997 | .94399 


| 33983 | 94049 
| 84011 | .94039 
| 84038 | .94029 


Sine |Cosin 
» BR557 | 94552 | 


82584 | 94542 | 


. 32612} .94533) ¢ 


82639 | 94523 | 
32667 | 94514 | 
82694 | 94504 | 
182722 | 94495 | 
382749 | 94485 
82777 |. 94476 | 
82804 | 94466 | 
- 82832 | 94457 
82859 | 94447 
. B2887 | 94438 
.82914) 94428 | 
.82942| 94418 
32969 | 94409 


.33024| 94390 
33051 } 94380 
33079 | .94370 
33106 | .94361 


33134 | 94351 
33161 | 94342 
33189 | 94332 | 
33216 | 94322 
| 88244! 94313 
| .33271 | 94303 
| 83298 94293 | 


. 383326 | .94284 
| .83353 | .94274 
. 83381 | 94264 


33408 | .94254 | 
33436 | 94245 
83463 | 94235 
33490 | 94225 
33518 | .94215 
33545 | 94206 
33573 | .94196 | 
33600 | 94186 
33627 | 94176 
33655 | 94167 


83682} 94157 | 
.33710| 94147 
| 88737 | 94137 
| 83764 | .94127 
| 83792 | 94118 
33819 94108 
33846 | .94098 | 
33874 | .94088 
33901 | 94078 
.33929 | .94068 


33956 | .94058 


84065 | .94019 


. 84093 | 94009 
.94120 | .93999 


34147 | 93989 | 
|| 84175 | 98979 
| 34202) 98969 


Cosin | Sine |: 
' 


x, 15° 
Sine | Cosin 
0 | .25882] 96593 | 
1 | .25910) .96585 | 
2 | .25938) 96578 | 
8 | .25966! .96570| 
4 | 25994) .96562| 
5 | .26022) .96555 
6 | .26050) 96547 | 
7 | .26079) .96540| 
8 | .26107) 96532} 
9 | 26135) 96524 
10 | .26163! 96517’ 
11 | .26191| 96509) 
12 | .26219) 96502 | 
HL 13 | .26247) 96494 | 
14 | .26275| .96486 | 
| | 15 | .26303] .96479 | 
16 | 26331) .96471 
17 | .26359| .96463 | 
. 18 |. 26387 | 96456 | 
We 19 | .26415} .96448| 
Ha 20 | .26443 06140) 
vi 21 | .26471| 96433; 
Vill 22 | 26500) .96425 | 
Bat 23 | 26528) 96417; 
. 24 | 26556) 96410, 
HE it 25 | .26584) .96402 
i 26 | .26612) .96394 | 
hil 27 | .26640) .96386 | 
28 | .26668 | 96379! 
29 | .26696) .96371 | 
80 | .26724| 96363) 
31 | .26752| .96355 | 
2 | 26780) .96347 | 
| 33 | .26808] 96340] 
fi 34 | .26836| .96332 | 
WW 35 | 26864] .96324 | 
i 36 | .26892) .96316 | 
Hh, 37 | £26920} .96308 | 
{ 38 | .26948| 96301 | 
Tt 39 | .26976| .96293 | 
ii 40 | .27004! . 96285 | 
41 | 27032) . 96277 | 
Bee: 2 | .27060) .96269 | 
aN 43 | .27088} .96261 | 
Bae il 44 | .27116| .96253 
45 | .27144| .96246} 
HEL 46 | .27172) . 96238 | 
\ 47 | .27200| .96230! 
WM) 48 | 27228} 96222) 
. 49 | 27256] .96214 | 
50 | .27284) .96206 
51 | .27312| .96198 
52 | 27340] 96190 
53 | 27368} .96182 
54 | 27396) .96174 
55 | .27424! .96166 
56 | 27452! .96158' 
57 | .27480) .96150 
58 | .27508) .96142 
59 | 27536) .96134 
60 | .27564! 96126; 
Cosin| Sine 
/ 
74° | 


73° 


Sine Cosin |, Sine |Cosin || 
. 29237 .95630 || .30902 | .95106 
29265 | .95622 |! .30929 | .95097 
29293 .95613 .30957 | .95088 
.29321 | 95605 .30985| 95079 
.29348 | .95596 |. .31012| .95070 
.29376| 95588 .31040| 95061 
29404) 95579 .31068| .95052 
29432] .95571) | 81095 | .95043 
29460! 95562) .31123)| 95033 
29487 | .95554|) 81151 | .95024 
.29515} .95545 | .31178]| .95015 
29543|-.95536 || .81206/.95006 
29571 | 95528! 31233) .94997 
. 29599 | -95519 |) .81261| .94988 
| .29626| .95511 |! .31289) .94979 
| 29654} .95502 || .31316' .94970 
.29682) 95493 |; 31344) 94961 
.29710| .95485 || .31372) .94952 
.29737|".95476 || 813899 | .94943 
.29765 | .95467 || .381427| 94933 
29793) .95459 | 81454 94924 
| .29821 | .95450!| .31482) .94915 
| 29849 | 95441 |) 81510! .94906 
.29876 | 95433 || .81537| .94897 
.29904) .95424|) 81565 | .94888 || 
.29982) 95415 |) .81593| .94878 
. 29960} .95407 || .31620/ .94869 
.29987| 95398 || .31648| .94860 
80015} .95389 || 31675! 94851 
80043 | 95380 || .31703! 94842 
80071 95372, .81730| .94832 
80098] .95363 | .31758| .94823 
.80126 | 95354)! .81786| .94814 
80154} .95345 |) .31813| .94805 
.80182| .95387 || .81841 | .94795 
.80209| 95328 || .81868] .94786 
. 80237 | .95319|| .31896| .94777 
.80265 | .95310|| .31923] .94768 
. 80292). 95301 || .81951| .94758 
. 80320) . 95293 || .81979| .94749 
. 80348} .95284 || .82006| .94'740 
.80376| .95275 || .82034] .94730 
.80403 | .95266 || .32061 | .94721 
. 80431 | .95257 || 32089! .94712 
80459 | 95248!) .32116| 94702 
80486 | .95240 || 32144] 94693 
.80514| 95281 || .32171| .94684 
80542! 95222 }| .82199| .94674 
.80570! .95213 || .82227) .94665 || 
.80597 | 95204 || .32254] .94656 
. 80625 | 95195 || .82282} 94646 
.80653 | .95186 || .82309| .94637 
| 80680] .95177 || .82337| .94627 
| .30708|.95168 | 82364] .94618 
.80736 |.95159 || .82892] .94609 
. 80763 | .95150 || .82419| .94599 
| .80791 | .95142)| .82447] 94590 
.80819 | .95133 || .382474| .93580 | | 
30846 |. 95124 || .82502! 94571 | 
30874 |.95115 || .32529) .94561 
.80902 |.95106 || 82557) 94552 
Cosin | Sine || Cosin} Sine | 
ee 71° i 


70° 


452 


, 


60 
59 


hes ig ia cen cameos 


» 


TABLE XXVII.—NATURAL SINES AND COSINES. 


gz 20° |_21°_ji_2ae 23° 94° 
Sine Cosin | Sine Cosin | Sine Cosin | Sine \Cosin |! Sine Cosin | 
0 | 34202). 93969 || 35837 | 93358)! .387461 92718 | _39073 | 92050 || .40674 .91355' 60 
1 | .34229/ 93959 |) 85864 93348) 37488 | .92707 || 39100! .92089 || .40700 .91343 59 
9 | 3425793949 || 35891 | .93837|| 37515 | .92697) 39127 | .92028 || .40727 .91331! 58 
3 | .34284| .93939|| .35918 93327 || .37542| .92686 ||'.39153| 92016 || 40753 .91319 57 
4 | 34311) .93929]|| 35945) .93316|| 87569) .92675 || .39180| .92005 || .40780 .91807 56 
5 | .34339)| .93919|| 85973) .93306'| 37595! .92664|| .89207| .91994 || .40806 .91295 55 
|| 6 | .34366| .93909 || .36000) 93295 '| .37622| .92653|| .39234 | .91982 || .40883|.91283 54 
7 | 34393] .93899 || .36027 |. 93285 37649 |. 92642|| .89260| .91971 || .40860 .91272 53 
. 8 | 34421 | 93889 || .36054! 93274) | .37676 | .92631|| .89287| .91959 || .40886 | .91260 52 
9 | .34448| .93879|| .36081 | 93264 | .37°703 | .92620|| .89314|.91948}| .40913! 91248 51 
10 | 34475] .93869|| .36108 | .93253)| .37730] .92609|| .89341 | 91936 || .40939|.91286 50 
| 11 | 34503 | .93859 86135) .93243]| 37757) .92598}| .89367| .91925]| 40966] 91224! 49 
| 12 | 34530| .93849|| .36162/ .93232|| .387784| .92587|| .89394| .91914|| .40992) 91212 48 
ie | 13 | .34557| 93839 || .36190] .93222 ‘37811 || 92576 || .89421 | .91902|| .41019| .91200) 47 
. 14 | 34584! .93829|| 86217) .93211 "37838 | .92565 || 89448] 91891 || .41045|.91188) 46 
| 45 | 84612) .93819|| .36244| .93201 | 37865 | .92554 .89474| 91879 || 41072] .91176| 45 
16 | .34639| .93809|| .36271 | .93190|| .87892/ .92543}| 89501 | .91868 || .41098| .91164 | 44 
| 17 | 34666} .93799|| .86298| .93180|} 37919! .92532|| .89528 | .91856 || .41125|.91152 43 
18 | 34694) .93789 | 86825] .93169 37946 | 92521 || 89555 91845 || .41151|.91140; 42 
. 19 | .34721|.93779|| .36352/ .93159)| .87973] .92510|| .89581 | 91833 || 41178) 91128) 41 
. 20 | 34748 93769 ‘36379 03118) .87999 | .92499|| .39608 | .91822/| .41204|.91116) 40 
| 21 | 34775 | .93759|| .86406| .93187!| .88026 | .92488 || .89635| .91810)| .41231}.91104) 39 
2 | 34803 | .93748 || 86434) .93127|| .88053 | .92477|| .39661| .91799|| .41257|.91092) 38 
23 | 34830] .93738 || .86461 | .93116|| .88080) .92466 || 89688 | 91787 || 41284) .91080 37 
24 | 84857! .93728 || .36488] .93106]| .88107 | .92455 || .89715| .91775 || .41310| 91068) 36 
| 25 | 84884! 93718 || 86515 | .93095)| .88134! .92444|| 39741 | 91764 | 41387) .91056 35 
| 26 | .84912| 93708 || .86542 .93084|| .88161 | .92432)| 89768] 91752 || .41363|.91044) 34 
27 | 34939 | 93698]! .86569| 93074 |; 88188] .92421 || .39795| .91741 || .41390) 91032! 33 
| 28 | 34966 | 93688 || .36596 | .93063)| .88215  .92410)| 39822} 91729 |] .41416| .91020 32 
29 | 34993 | 93677 || .86623/ 93052]; 28241! .92899|| 89848} .91718|| .41443] 91008 | 31 
| 30 | 35021 | .93667 -36650| .93042) eng: .89875 | 91706 || .41469) .90996 30 
31 | .35048| .9365? || .86677 | 93031 || .38295| 92377]! 39902) .91694 || .41496 90984! 2 
82 | .35075| .93647 || "36704 -93020|! .88322 ees .89928| .91683 || .41522| 90972) 2 
33 | .35102! .93637 || .86731! .93010|| .88349 | .92355|| 89955) .91671 || 41549} 90960) 27 
. | 94 | 351301. 93626)! .86758| 929991! 88376! .92343 || .39982) .91660}| .41575| .90948) 26 
35 | .35157 | .93616 || 86785 | .92988'| .88403| .92332]) .40008|.91648|| .41602/ .90936 | 25 
36 | 35184! :93606 | 36812 .92978 | .38430! .92321 || .40035| 91636 || .41628].90924' 24 
37 | .35211| 93596 | 86839 .92967 | .38456) .92310|| .40062) .91625|| .41655).90911| 23 
38.| 85239 .93585 | .86867 | 92956 | .88483 | .92299|| .40088| 91618 || .41681 | .90899| 22 
39 | .35266|.93575 | .36894 | .92945 | 88510) .92287|| .40115| .91601 || .41707| 90887! 21 
40 | .85293| .93565_ 36921 |. 2935 || .88537'| 92276 || .40141 | .91590|| .41784 | .90875 | 20 
‘41 | 85320) .93555 | .86948 | .92924 || .88564| 92265 || .40168] .91578 || .41760) .90863) 19 
42 | 35347| 93544 | 36975 | .92913|| _88591| _92254/| .40195) .91566 |) .41787| 90851! 18 
| 43 | .35375|.93534 | .87002/ .92902!| .88617) 92243] .40221 | .91555 || .41813) 90839 17 
| 44 | 35402! _93524'| .3702 | 92892 || 38644) 92231 || 40248] 91543 | .41840/ .90826| 16 
45 | 35429! .93514 | .87056| .92881 || .88671 | .92220|| .40275| .91531 || .41866| .90814| 15 
. \ 46 | 35456! .93503.| .87083! .92870 || .88698 | .922 309 | "40301 .91519|| .41892/.90802) 14 
7 | 35484 93493 | 87110 .92859|| .38725' 92 2198 | .40828 | .91508 || .41919| .90790| 18 
48 | .35511|.93483 | .87137| .92849|| .88752 | .92186 || .40855 | .91496|| .41945) 90778) 12 
49 | 35538) .93472 | .37164!.92838 |} .38778| .92175|| .40881 | .91484 || .41972|.90766| 11 
50 | .35565 | .98462 | .87191}.92827 |! .88805).92164|| .40408 .91472|| .41998] .90753) 10 
51 | 35592! .93452 | 87218] 92816]! .38882! .92152/| .40434 .91461 || .42024].90741} 9 
! 52 | 35619! .93441 | .87245} 92805 || .88859 | 92141 || .40461 .91449 || .42051| .90729| 8 
53 | .35647| .93431 || .87272| .92794 | 38886 | 02150 40488 .91437|| .42077|.90717| 7 
| 54 | 35674) 93420 | .37299| .92784|| .88912' 92119) .40514 .91425]| .42104|.90704| 6 
55 | 35701! 93410 | .37326| .92773|' .38939!.92107!| .40541 .91414)/ .42130|.90692] 5 
| 56 | 35728} .93400 | .87353| .92762!) 88966 .92096 || .40567 .91402 || .42156|.90680| 4 
7 | .85755| .93389 | .87380| 92751 |) .88993'.92085 |) 40594 .91390 || .42183] 90668] 3 
58 | 357821 .93379 | .87407 "92740 39020 .92073 || .40621 .91378|| .42209!.90655; 2 
59 | 35810! .93368 | .87434).92729|) 39046’ .92062||.40647 .91366 || .42235/.90643| 1 
60 | 85837 93358 | .37461|.92718) 39073 .92050 | 40674 91355 || 42262) -90631| 0 
| : Cosin | Sine || Cosin | Sine | Cosin | Sine | Cosin Sine Cosin | Sine 
. 69° | 682 67> ~—s«d||=s«G 65° 


458 


TABLE XXVII.—NATURAL 


SINES AND COSINES. 


iz | 


62° 


Co OT = COC 


||| a a ee Cy aaa | ue ae” 
Sine |Cosin || Sine |Cosin || Sine |Cosin | Sine |Cosin | Sine |Cosin 
0 | .42262 | .90631 || .43837|.89879|| .45399 | 89101 |) .46947 | .88295 | .48481 | 87462] 60 
1 | .42288/ .90618 || .48863) .89867 || .45425 | .89037 || .46973 | .88281 || .48506 | .87448) 59 
2 | .42315 | .90606 || .48889| .89854 || .45451| 89074 || .46999 | .88267 || .48582 | .87434) 58 
3 | .42341 | 90594 || 48916} 89841 |) .45477) .89061 || .47024| 88254 || .48557 | .87420|. 57 
4 | .42367 | .90582 || .48942] .89828 |} .45503| 89048 || .47050| .88240 || .48583 | 87406) 5 
5 | .42394 | .90569 || 48968 | .89816 || .45529 | .89035 || .4'7076 | .88226 || .48608 | .87391} 55 
6 | .42420) .90557 || .48994| .89803 || .45554| 89021 }| .47101 | .88213 || .48634) .87377| 54 
7 | .42446 ; 90545 |) .44020) .89790 || .45580 | .89008 || .47127| 88199 || .48659 | .87363! 53 
8 | .42473 | 90532 |) .44046 | 89777 || 45606 | £88995 |) .47153 | .88185 || .48684 | .87349 | 52 
9 | 42499) .90520 |) .4407'2| 89764 || .45632] .88981 || .47178 | .88172 || .48710| .87335| 51 
10 | .42525/.90507 || .44098| 89752), .45658| .88968 || .47204 | .88158 || .48735 | .87321 | 40 
11 | .42552! .90495 |) .44124) .89739 |) .45684| .88955 || .47229| .88144 || .48761 | .87306) 49 
| 12 | .42578| 90483 |] .44151/ .89726]|| 45710] .88942 || .47255 | .88130 || .48786| .87292| 48 
) 13 | .42604) 90470 || .44177' .89713 || .45736 | .88928 || .47281] .88117 || .48811 | 87278} 47 
14 | .42631 | 90458 || 44203; .89700 || .45762] .88915 || .47806 | .88103 || .48837 | .87264| 46 
15 | .42657 | .90446 || .44229 | .89687'|| .45787 | .88902 || .47332] 88089 || .488621.87250) 45 
16 | .42683| 90433 |) 44255) .89674 || .45813].88888 |) .47358] .88075 || .48888|.87235} 44 | 
‘ 17 | .42709) .90421 || .44281| .89662}| .45839 | .68875 || .47383 | 88062 || .48913! 87221 | 43 
18 | .42736 ;.90403 || .44307} .89649 || .45865 | .88862 || .47409|.88048 || .48938|.87207| 42 
19 | .42762| .90396 || .44333! .89636 || .45891 | 88848 || .47434 | .88034 |) .48964 -87193 | 41 
20 | .42788 | 90383 | 44359 | 89623 || .45917 | .88835 || .47460 | 88020 || .48989 | .87178| 40 
21 | 42815] .90371 || .44885).89610|| .45942| .88822 || 47486 |. 88006 || 49014 .87164| 39 
2 | 42841) 90358 || .44411| .89597 || .45968| 88808 || .47511| .87993 || .49040|.87150] 38 
23 | .42867 | 90346 |) 44437! 89584 || .45994) .88795 || .47537 | 87979 || .49065 | .87136| 37 
24 | 42894) 90334 || 44464) 89571 || .46020|.88782!) .47562|.87965 || .49090] .87121| 36 
i 25 | .42920) .90321 || .44490] .89558 || .46046| .88768 || .47588 | 87951 || .49116]| .87107 | 35 
i 26 | .42946| .90309 || .44516) .89545 || .46072|.88755 || .47614| .87937 || .49141|.87093| 34 
27 | 42972} 90296 |) 44542) 89532 || .46097 | .88741 || .47639 | 87923 || .49166| .87079| 3: 
Hii 28 | .42999 | 90284 || 44563! .89519 || .46123| .88728 || .47665 | .87909 || .49192] .87064 | 32 
| 29 | .43025 | 9271 |) .44594) 89506 || .46149| .887'15 |} .47690 | 87896 || .49217) 87050} 31 
4 30 | .48051| .90259 |! .44620) .89493 || .46175| .88701 || .47716 |. 87882 || .49242) .87036} 30 
a) 31 | .48077) .90246 | .44646] .89480|| .46201 | .88688 || .47741 | .87868 || .49268}.87021] 29 
Hy 2 |.43104/ 90233 || .44672) .89467|| .46226 | 88674 |! .47767 | .87854 || .49293] .87007| 28 
Pit 33 | 43130) .90221 || .44698} .89454]] .46252| .88661 || .47793 | .87840 || .49318} .86993} 27 
i 34 | 43156} .90203 || .44724) .89441|| .46278| 88647 || .47818 | 87826 || .49344| .86978| 2 
iM 35 | .48182) .90195 || 44750} .89428 || .46304| .88634 || .47844| .87812|| .49369] 86964) 25 
i 36 | 43209) .90183 || .447'76) .89415]| .46330| .88620|| .47869 |.87798 || .49394| .86949|-2 
Wh 37 | 48235) 90171 || 44802) .89402]) .46355| .88607 || .47895 | 87784 || .49419| .86935| 23 
HI 88 | .43261 | .90153 || .44828) .89389]| .46381 | .88593 || .47920 | .877'70 || .49445| .86921 | 22 
39 | .43287) 90146 | 44854 .89376 || .46407 | .88580)| .47946 | .87756 || .49470} 86906} 21 
40 | .43313 | .90133 | .44880) .89363 || .46433 .88566 || .47971 | 87743 || .49495| .86892| 2 
41.| .48340] 90120 | .44906| .89350 |! .46458 | .88553}| .4'7997’| .87729 || .49521].86878] 19 
42 | .43366 90103 | 44932) .89337|| .46484! .88539 || .48022) .87715 || .49546| .86863| 18 
43 | .43392 | 90095 || .44958) .80324|| .46510) .88526 || .48048) .87701 || 49571} 86849} 17 
44 | .48418) 90052 || 44984) .89311]| .46536 | .88512]| .48073 | .87687 || .49596| .86834| 16 
45 | .43445} 9000 || .45010; .89298 || .46561 | .88499|| .48099! .87673 || .49622] .86820| 15 
46 | .48471) .90057 || 45036) .89285 || .46587 | .88485 | -48124 | .87659 || 49647) .86805| 14 
| 47 | 48497) 90045 |) 45062) .89272 |) .46613 | .88472|| .48150|.87645 || .49672| 86791] 13 
' 48 | 43523] .90032 || .45088| .89259]| .46639 | .88458]| .48175 |.87631 || .49697 | .8677'7| 12 
49 | .43549 90019}; 45114) .89245 |) .46664' .88445 |! .48201 | .87617|| .49723] 86762} 11 
50 | .48575/ 90007 || .45140] .89232) .46690|.88431 || .48226 |.87603 || .49748] .86748| 10 
51 | 43602 89994 | 45166 | .89219|' .46716 | .88417| | .48252].87589 || .49773} .86733 
52 | .43628] .89981 |) .45192| .89205| .46742| 8840! .48277] .87575 || .49798| .86719 
53 | .43654/ .89968 |) .45218] .89193|! .46757 | .68399|| .48303|.87561 || .49824| .86704 
54 | 43680} .89956 || .45243|.89180), 46793 | .88377'|| .48328 |.87546 || .49849 | .86690 
55 | .43706 | .89943 || .45269|.89167|' .46819| .88363|| .48354|.87582|| .49874] .86675 
56 | .48733] .89930 || 45295} .89153|) .46844 | .88349| | .48379|.87518|| .49899| .86661 
57 | .43759| .89918 || .45321 | 89140} .46870|.88336 | |’.48405 |.87504/| .49924] 86646 | 
58 | .43785) .89905 || .45347! .89127 || .46896 | .88322 || .48430 |.87490 || .49950| .86632| 2 
| 59 | .438811) 89892 || .45373}.89114|| .46921 | 88308 || .48456 | 87476 || .49975|.86617) 1 
| 60 | 43837 | .89879 || .45399| 89101 || -46947 | 88295 |! .48481 | 87462) | .50000) .86603| 0 
; Reco Sine || Cosin| Sine | Cosin | Sine |, Cosin } Sine p eae Sine f 
ee ee ~ \ | = 
| 64° | 63° 61° 60° 


TABLiC 


XAVi 


© | .50000 | :86603 | 
| 50025 | .86588 | 
| 50050} .86573 | 
3 | .50076 | .86559 | 
50101 | .86544 | 
.50126 | .86530 | 
| .50151|.86515 
.50176 | .86501 | 
| 50201 | .86486 

| 50227 | 86471 
| 50252] .86457 


2 | 50302 | .86427 
50327 | .86413 
| .50352} .86398 

| 50377 | .86384 
50403 | .86369 
-50428| .86354 
50453 | 86340 | 


ne | 
fek e at ae 
NQOUBWWRH COOVIMWOILWWeH 


(@ 2) 


rw 


.50503 | .86310 
21 | .50528 | .86295 
22 | .50553 | .86281 | 
23 | 50578 | .86266 


Lav) 
Bag 


1D 2 
v 


50628 |. 86237 
26 | .50654 | .86222 
97 | 50679 .86207 
28 | .50704| .86192| 
29 | 50729) .86178 | 
30 | .50754|.86163 
31 | .50779| .86148 | 
32 | .50804|.86133 | 
| .50829|.86119 
34 | 50854 86104 
35 | .50879|.86089 | 


| 
3G | 50004| 86074 
| 
| 
i 
| 
| 


ri 
oO 


oe 
co 


~) 


) 
2 


37 | .50929 | .86059 
38 | 509541 86045 
| 00979 | .86030 
40 | 51004! .86015. 
41 | .51029) .86000 | 
42 | 51054|.85983 | 
43 | 51079 |.85970 
44 | 511041" 85956 
45 | 51129) .85941 | 
| 46 | 51154) .85926 
; 47 | .51179|.85911 | 
48 | 51204! .85896 | 
49 | 51229). 85881 | 
50 | .51254 | .85866 
51 | .51279|.85851. 
52 | 51304 | .85836 | 
53 | .51329) 85821 | 
| 54 | .51354|. 85806 
55 | 51379) 85792 
56 | 51404] .85777 
BY | 51429). 85762 
58 | 51454! 85747 | 
59 | 51479) 85732 
60 | 51504! 85717 


oo 
oc 


- 50277 | 86442 


| 50478 | 86325 


| 50603 | .86251 


| .51823 
.51852 


| .52026 


opel | ae ae 
Sine |Cosin | 
.51504 
51529 | 
51554 | 
.51579 
| 51604 
.51628 
| 51653 
.51678 
.51703 
51728 
51753 | 


51778 | 
51803 | 


51877 
.51902 
.01927 
.51952 
51977 
52002 | 


52051 
52076 
.52101 
.52126 
02151 
.O2175 
.52200 | 
52225 


92250 


52275 
02299 
52324 
52349 
52374 
.52399 

52428 
52448 
02473 
52498 
02522 
S247 
52072 


le 


-02597 | 
.52621 
.52646 
.52671 
. 52696 
.52720 
.52745 
20% 

.52794 
.52819 
52844 
.52869 
.52893 
.52918 
.52943 
.52967 | 
. 52992 


Cosin 


85717 || .52992' 84805 

.85702 || .53017) .84789 

85687 || .53041} 84774 
.85672 || 58066) .64759 
85657 || .53091 | .84743 

85642 | .53115) .84728 
.85627 |! .58140) .84712 
85612 || .58164| .84697 

85597 || .58189| .84681 

85582 || .53214| .84666 
85567 || .532388 | .84650 
85551 || 58263} .84635 | 
85536 || .538288!.84619 
85521 || .53312 | .84604 
85506 || .53337 | .84588 
85491 || .53361 | 
85476 || .53386 | 84557 | 
.85461 ||-.53411) .84542 | 
85446 || 53435 | 84526 
85431 || .58460) 84511 | 
85416 || .53484) .84495 | 


.85401 || .53509 | .84480, 
85385 || .53534) 84464 
.85370 | .53558 | 84448 
85355 || .53583 | .84433 | 
.85340 || .53607 | 
.85325 || .53632 | 84402 
.85310 || .53656 | 84386 
.85294'| .53681 | 84370 

.85279 | 53705 | 84355 

see .53730 | 84339 

85249 || 53754 | 84324 
.85234 || .537'79 | .84308 
85218 || .53804 | 84292 
.85203 || 53828 | 84277 
85188 || .538853 | .84261 
.85173 || .53877 | 84245 
.85157 || .53902} .84230 
.85142 || .53926 | 84214 
85127 || .53951 | 84198 
.85112 | 53975! 84182 
.85096 || .54000 
85081 || .54024| 84151 | 
.85066 || .54049 
.85051 || .54073| 84120 
.85035 || .54097|.84104 | 
.85020 || .54122| 84088 
.85005 | .54146}| .84072: 
.84989 || 54171} .84057 | 
84974 | 54195) .84041 || 
84959 || 54220) .84025 | 


84943 '| 54244! 84009 | 
84928 || 54269) .83994 | 
84913 || 54293 | 
84897 || 54317! .83962 | 
84882 | 54342 |.83946| 
(84866 || .54366 | .83930) 
84851 || 54391! 83915 | 
84836 | if 
84820 | .54440' 83883 | 
84805 | 54464) .83867 


Sine |Cosin 


84575 


84417 


,84167 | 


84135 | 


.83978 


.54415! 83899 


Sine |Cosin| Sine || Cosin | 


| Cosin | Sine | 
| 


i Soe. | 


33° 


IL—NATURAL SINES AND COSINES. 


34° 


54854 | 
'54878 | 
54902 
4927 
54951 | 


AQT | 
54999 | 
55024 
55048 | 
(55072 | 
55097 | 
"55121 | 
155145 | 
55169 
55194 | 


.55218 | 
55242 | 


£5266 | 


55291 | 
55315 | 
55339 | 
55363 | 
55388 | 
55412 
.55436 | 
55460 | 
55484 | 
.55509 
55533 | 
LBBBDT | 
55581 | 
55605 | 
55630 
B5GD4 | 
55678 | 
55702 
55726 | 
.BDT5O 
B5G75 | 
5S799 
.5b823 
55847 | 
55871 
55895 
55919 | 


Sine |Cosin 
.54464 
.54488 
.54513 
.54537 | 
.54561 
| 54586 
.54610 
54635 | 
| 54659 
.54683 
.54708 
54732 | 
.54756 
54781 
54805 | 
.54829 | 


. 83645 || .56256 
83629 || .56280 
83618 || .56805 
83597 || .56329 
83581 || .56353 
83565 || 563877 
.83549 || .56401 | . 


83533 || 56425 
(83517 ||-.56449 
| 83501 | 56473 
83485 | .56497 
(83469 || .56521 
(83453 || .56545 
83437 || .56569 
(83421 || .56593 
-83405 || .56617 
.88389 || .56641 | 
83373 || .56665 
(88356 ||. .56689 
.83340 || 56713 
88324 || .56736 
.83308 || .56760 
83292 || .56784 | 
88276 || .56808 | 
“$3260 | .56882 
83244 || 56856 
88228 | .56880 
88212 || 56904 
83195 | 
83179 
83163 || .56976! 
.83147 | 
83131 || ..57024| 
(88115 || 57047 | 
88098 .| 57071 
-83082 | 57095 
-§3066 |; .57119). 
83050 
| 83084 ||..57167 
88017 | .57191 
83001 || 57215 
-82985 ||57238 
82969 
~ 82953 | 1-. 57286 
“82936 |. 
82920 | 
82904 
Sine || C 


.56088 


.56160 | 


| 


56928 | 
.56952 


57000 | 


57143 


pt ard 
.57 262 


Sine |Cosin 

.83867 || 55919 | 
83851 || 55943 | 
83835 || .55968 
83819 || .55992 | 
.83804|! .56016). 
.83788 || .56040 | 
83772 || 56064 
83756 | 
83740 || .561121. 
.83724 || 56136 | 
.837u8 


.83692|| .56184 
83676 | | .56208 
83660] | .56232 


82904) 60 


82887 | 
82871 
82855 
82839 


RVQOW 


-OROKG + 


82806 


.82790 5: 


eorry 
e 


82757 
82741 
82724 
82708 


82692 | ¢ 


82675 | 
82659 
82643 
82626 
82610 
82593 


82577 


| 
82561 
82544 


.82028 * 


82511 
82495 
82478 
82462 
82446 
82429 


82413 § 


82396 
82380 
82363 
82347 
82330) 


82314! 2 


82297 | 


82281 


82264 | 2 


82248 


5008 | 
82214 | 
82198 | 
82181 | 
82165 | 
82148 
82132 
82115 | 
“82098 
82082 
82065 | 
82048 | 
82082 | 
82015 | 
81999 | 
81982 | 
“81965 


81949 
| 81932 
58) 81915 | 


sae | mee 


56° 


455 


Sine | ; 


TABLE XXVII.—NATURAL SINES AND COSINES., 


35° 


~ 


Sine Cosin 


| 
| 


ee 


38° 


| 
| 
| 


| 


| 36° 
Sine |Cosin 
58779. 80902 | 
| 58802} .80885 
| .58826 |. 80867 
| .58849 |. 80850 | 
| 58873 | .80833 
58896 | .80816 
| 58920 | .80799 | 
| 58943 | 80782 | 
.58967 | 80765 
58990 | .80748 
.59014 | .80730 
.59037 | 80713 
59061 | .80696 
.59084 |. 80679 | 
59108 | 80662 
59131 | 80644 
59154! 80627 | 
.59178 | .80610 
.59201 | .80593 
59225 | 80576 
59248} .80558 
.59272 | 80541 
59295 |. 80524 
.59318} .80507 
-59342) .80489 
59365 | .80472 
59389 | 80455 
59412) 80438 
59436} .80420 
.59459 | .80403 
. 59482! 80386 
59506 | .80368 
.59529 | .80351 
.59552 | 80334 
.59576 | .80316 
59599} .80299 
| .59622/ 80282 
. 59646 | . 80264 
.59669 | 80247 ; 
| .59693 | 80230 | 
| 59716 | 80212, 
.59739 | .80195 
.59763 | .80178 | 
.59786 | .80160 | 
| .59809/ .80143 
.59832 | .80125 || 
59856 | .80108 | 
.59879 | .80091 
.59902 | .80073 | 
59926 | 80056 | 
.59949 -80038 | 
59972} 80021 | 
.59995! 80003 | 
60019} .79986 
| .60042 | .7'9968 
| 60065 | .79951 
60089 | ..79934. 
| .60112).79916 
| .60135 | .79899 
.60158 | .79881 
60182) .79864 
|; Cosin | Sine | 


Sine \Cosin | 
.60182 | .79864 | 
60205 | .79846 | 
. 60228 | .79829 
60251 | .79811 | 
60274 | .79793 

60298 |. 79776 

. 60321 | 79758 | 
60344 | 79741 | 
.60367 | .79723 | 
. 60390! .'79706 | 
60414 | 79688 | 
.60437 | 79671 | 
60460 .79653 
.60483 | .79635 
.60506 | .79618 
.60529 | .79600 | 
. 60553 | .'79583 | 
60576 | . 79565 | 
60599 | 79547 


61566 | 
61589 | 


60622 | 79580 
.60645 | .79512 
60668 | 79494 
60691 | 79477 
60714 | .79459 
60738 | .79441 
60761 | 79424 
60784 | 79406 | 
.60807 | .79388 
60830 | 79871 | 
60853 | .79353 | 
.60876 | .79335 
.60899 | 79318 
.60922 | 79300 
60945 | 79282 | 
60968 | 79264 | 
60991 | 79247 
61015 | 79229 | 
61038 | 79211 | 
61061 | 79193 | 
.61084 | 79176 | 
61107 | 79158, 
.61130| 79140! 
61153 | 79122 | 
61176 |. 
61199 | 
61222 | 
61245 | 
61268 | 
61291 
61314 
61337 
61360) 
61383 | 
.61406 | 
61429 
61451 | 
61474 
61497 | 
61520 


79105 || 
79087 | 
79069 | 
79051 
. (9033 
79016 || 
78998 | | 
78980 | | 
| 

. 78962 | 
78944 || 
. 78926 | | 
. 78908 | | 
78891 | | 
78873 | | 
. 78855 | | 
78837 | | 
61543 | 78819 | 
.61566 | . 78801 
Cosin | Sine 


|| 


. 78801 | | 
T8783 | | 


Wee ae 


Sine |Cosin || Sine 


62932 
62955 | 


Cosin 


fd 
. T7696) 5$ 


oO 


15 


pipded 


oe 


0 | 57358! 81915! 
.57B81 : .81899 
2 57405. .81882 
3 | .57429' 81865 | 
4 | .57453' 81848 
5 | 57477} 81832) 
6  .57501! .81815/ 
7 | 57524) 81798 
8 .57548) .81782 
9 | .57572| 81765 
10 | .57596) .81748 
11 | 57619! .81731 
12 | .57643|.81714| 
13 | .57667| .81698 
14 | .57691|.81681 
15 | 57715! .81664 
16 | 57738) 81647 | 
17 | 57762) 81631 
18 | 57786) .81614| 
. 19 | 57810! .81597| 
nh 20 | .57833! .81580 
a 21 | 57857! 81563 
Hirata 22 | 57881) .81546 
Be 23 | 57904) 81530) 
i 24 | 57928) 81513 | 
25 | 57952) 81496 :. 
26 | .57976| .81479| 
27 | .57999| .81462' 
28 | 58023! 81445 | 
29 | 58047) 81428 | 
i 80 | .58070.81412) 
31 | 58094) 81395 | 
: 82 | 58118) 81378) 
33 | 58141) .81361 | 
i 34 | 58165) .81344, 
it 85 | 58189) .81327 | 
Ny 36 | 58212] 81310. 
87 | .58236) .81293 | 
| 38 | .58260/ .81276 | 
at 39 | 58283 | 81259. 
i 40 | .58307/.81242 
pH 41 " 58380) 812951 
42 | 58354! .81208 | 
| 43 | 58378) 81191 | 
44 | 58401) .81174' 
45 | 58425) .81157| 
46 | 58449, 81140 / 
4 | 58472; 81193} 
48 | 58496) .81106 | 
49 | 58519} .81089 | 
50 | .58543}.81072 | 
51 | .58567!.81055 
52 | .58590) .81038 | 
53 58614) 81021, 
54 | 58637) 81004 | 
55 | 58661 | 80987 
56 | 58684 | .80970 || 
57 | 58708] 80953 
58 | .58731/ .80936 
59 .58755| .80919 
60 | .58779! .80902 | 
: 'Cosin | Sine 
| = 


53°C 


52° 


.61612 | .78765 || .629771 .7%678| 58 
.61635 | .78747 || .63000' .77660') 57 
.61658 | .78729 | .63022 .77641 | 56 
.61681 | 78711 || 68045 | .77623| 55 
.61704 | .78694 || .63068' 77605) 54 
.61726 | .78676 || .63090' .'77586! 53 
.61749| . 78658 || 63113! .77568! 52 
.61772| .78640 || .63135 °.7'7550! 51 
61795 | .78622 | 63158) .77581 50 
61818] 78604 | .63180! 77513} 40 
.61841 | 78586 || .63203!.77494| 48 
61864 | 78568 || .63225 .7'7476| 47 
.61887 | 78550 || .63248 77458] 46 
61909 | 78582 || 68271 77439) 45 
.61932 | 78514 || .63293'.7'7421| 44 
.61955 | .78496 || .63316 .'77402) 43 
.61978 | 78478 || 63338 .7'7384| 42 
.62001 | .78460|| .63361 .77366) 41 
62024) .78442 || 63383 .77347) 40 

"62046| 78424 | .63406 | .7'7329 39 

| 62069 |. 78405 || .63428 77310) 88 

| -62092| .78387 |) .63451 °.77292 37 
62115 | .78369 | 63473! _77273| 36 
.62188) .78351 || .63496 | .77255' 35 

| 62160-78333 || |63518 | 77236) B4 

| 62183 | .’78315 |) .63540'.77218| 33 
.62206 | 78297 || .63563 | 77199) 32 
.62229] .78279 || .63585/ .77181] 31 

| -62251 78261 .63608 | .7'7162| 30 

| 62274 | 78243 |] 63630! .7'7144| 29 
62297 | .78225 || .63653 | .777125 | 28 
. 62320 | .78206 || .63675 | .77107 | 27 
. 62842) 78188 || .63698 | .7’7088 | 26 
. 62365 | .78170 || .63720! .77070| 25 

| .62888 | .78152|| .63742| .7'7051| 24 
62411 | 78184 | .63765| 77033 | 23 
.62433 | 78116 || .63787|.77014/ 22 
.62456| .'78098 || .63810! .76996| 21 
.62479 | .78079 | .63832) .76977) 20 
.62502 | .78061 || . 63854 76959 | 19 
. 62524 | .78043 || .6387'7| 76940) 18 
.62547 | .78025 | .63899 | 76921 | 17 
.62570 | 78007 || .63922! 76903 16 
.62592 | .77988 | .63944| .76884/ 15 
.62615| .77970|| .63966 |. 76866) 14 
62638 | .77952 | .63989 | 76847 13 
.62660 | .77934 | .64011|.76828) 12 
.62683 | .7'7916 || .64033| 76810) 11 | 
62706 | .77897,| .64056 | .76791 | 10 | 
62728 | .77879|| 64078! "6772! 9 
62751 | 77861 || 64100) .76754) 8 
62774 | 77843 || 64123) .76735| 7 
62796 | .77824)| .64145|. 76717) 6 
.62819|.7'7806. | .64167|.76698| 5 
.62842) .'77788 || .64190'.76679! 4 
62864 | .'7769/| 64212 .76661! 3 
62887 | .77'751|| 64234-76642; 2 
.62909 | .77733)| .64256 16628 | 1 
.62932|.7'7715 || .64279' 76604; 0 

Cosin | Sine | Cosin | Sine 

51° 50° 


TABLE XXVIL—NATURAL SINES AND COSINES. 


: 
. 
a 
ie]. 40e.|j__ 41° 42° ate aa4 
7 || ae j a hs Gere re LS Soa Pe SS / 
| Sine |Cosin |; Sine |Cosin | Sine |Cosin|| Sine .Cosin || Sine Cosin|} 
| $70 | 64279 76604 |) 65600 |. 75471, | .66913} 74314 |) .68200 |.73150 | C9466 .71934' 60 
| 1 | 64301 |. 76586 ,| .65628|.75452 | .66935] 74295 || 68221 | .78116 | .COIS7 71914) 59 
9 | 6432376567 || .65650|.75433 || .66956 | .174276 || .68242 | 73096 || 69303. T1594! 58 
3 | 64346] .76548 || .65672).75414 | 66978 74256 || .68204| .738076 || 69529 71873, 57 
| 4 | 164368! .76530 || .65694 | .'75395 || 66999] .'74237 || .68285 | 78056 || .69549 .71853) 56 
5 | .64390|. 76511 || .65716| 75375 || .67021) 74217 || .68306 | 730364) 69570 71833) 55 
G | 64412! .76492)| .65738 | 75356 | 67043) 74198 .68327 | .73016 || .69591 | .71813) 54 
. 7 | 64435 |. 76473 || .65759|.75337 || .67064) .74178)| .68349 | 72996 69612 .71792) 53 
8 | 64457-76455 | .65781|.75318 || .67086 | .74159 || .68370 | 72976 || 69683 | 712, 52 
. 9 | 64179 | 76436 || .65803}.75299 |) .67107  .74139}] 68391 |.72957 || 69654! 71752, 51 
10 | “ake Nonna 65825 | .75280 || .67129 | .74120]| .68412 | .72937 69675 1732, 50 
11 | .64524) .76398 || .65847 75251 || .67151| .74100]| .68434 | .'72917 || .69696' .7 ml 49 
. 2 | 64546 | .76380 || .65869 |. 75241 || .67172) .74080}] 68455 | .72897 69717 | .71691| 48 
13 | 61568 .76361 || 65891) .75222 || 67194) .74061 || 68476) 72877 69737 | .71671| 47 
|| 1£ | 64590) 76342)/ 65913 |.75203 | “67215! .74041]} .68497 | .72857 || .69758 | 71650, 46 
|| .15 | 64612) .76323 | -65935 | .75184 | "67237 | .74022|| .68518 | .72837' || .6977'9 |.71630) 45 
16 | .64635' .76304/| .65956 "75165 || .67258| .74002|| .68539| .72817 || .69800|.71610) 44 
17 64057 76236 | .65978|.75146 || .67280| .73983|| .68561|.72797 || .69821 | 71590) 43 
: 18 },.64679 | .76267 | .63000 | .75126 | .67301| ..73963 || .68582 | 72777 || .69842 71569! 42 
19 | 64701) .76243 | .66022}.75107 || .67323) .73944)| .68608 T2757 || 69862 | .71549| 41 
: 20 pai eae | .66044 | .75088 | 67344 73924 || .68624| .72737 || .69883 | .71529| 40 
: 21 | .61746' .76210'| .66066 | .75059 | 67366) .739041| .68645| .72717 || :69904.| 71508) 39 
22 | 64768 .76192 | .63038|.75050 || .67387| . 78885 || .68666 | . 72697 | .69925 | .71488) 38 
i 23 | 64790, .76173 | 65109]. 75030 | .67409) . 78805 68688 | 72677 || .69946) 71468) 37 
. 54 | 64812! 76154 | .63131|.75011 || .67430] .73846 || .68709] .72657 || .69966 | .71447) 36 
25.6183) 76135 | .63153).74092 || .67452) .73826|| 68730 72637 | .69987 | .71427| 35 
25 | 64856 .76116 | 65175) .74973 | .67473} .78806)| 68751 72617 || .70008 | .71407| 34 
97 | 64873-76097 ,| 66197 |. 7.1953 | .67495| .73787|| .68772 | .72597 || .70029 | 71386) 33 
98 | .64901|.76078 | .65218|.74934 || 67516] .73767]| .68793 72577 || .70049| 71366) 32 
| 29 | 64923) 76059 | .66240|.74915 | .67538| 73747 |) 63814] .72557 | 70070) 71345) 31 
| 30 | 64945). 76041 | . 66262 | .74896 || .67559 23728 || .68835 | .'72537 || .70091 | :713825) 30 
31 | 64967) .76022 | .66284 74876 || £67580! .737038 || .63857 72517|| 70112! .'71805| 29 
39 | 64939 .76003 | 63306 |.74857 | .67602| 73633 || .68878|.72497 || 70132) .71284) 28 
| 33 | 6501173934 | .66327|.74838 | .67523] .73669|| .63899].72477 | .70153 | 71264) 27 
34 | 65033-75935 | .66349). 74313 | 67645} 73649 || 68920) 72457 -70174 | .'71243| 26 
35 | 65055! .75946 | .63371 |.74799 || .67686| .73629]| .68941 | .72437 |) .70195 (1228) 35 
: 36 | 65077) 75927 || 65393 |.74780 | .67638) .73610|| .68962 72417 | .70215| 71203) 24 
37 | 65100, 73903. | .68414|.74760 || 67709) .73590|| .68983] .72397 || 70236) 71182) 23 
98 | 65122. 75889 || .66436|. 74741 || .67730! 73570 || .69004| 72377 || . 70257 71162) 22 
39 | 65144) .75870 | .60453 |. 74722 | 67752] 78551 |) .69025 | 72357 |) 70277 .71141| 21 
40 preety be .66480 74003 || 67773 | «73531 || .69046 | .72387 || .'70298 | .71121) 20 
41 | 65188! 75832 66501 | .74633 | .67795) . 73511 69067 | 72317 || :70319| .71100| 19 
2 | 65210! .75813|| .66523|.74654 | 67816) 73491 || 69088 72297 |; 40339 71080) 18 
43 | .65232|.75794 | .66545|. 74644 | 67837) .73472|| .69109].72277 || 70860 71059] 17 
{ 44 | 65254! .75775 | .66566 | .74625 | 67859) .73452)| .69130) .7% 7\1,.70881 | 71039} 16 
45 | .65276, .75756 | .66588| 74605 | .67880) .73432)| .69151).72 | 704011 .71019) 15 
46 | 65298-75738 | .66610| 74535 | .67901! .73413|| .69172|.72216 || .70422 70998) 14 
| 47 | .65320; .75719 | .66632) .74567 || .67923| .73393 || .69193| 72196 | 70443! 70978, 13 
| 48 | 65342, .75700 | .66653 | 74548 || .67944) .73373|| .69214|.72176|| .70463 70957) 12 
| 49 | 65364! .75680 | .66675 | .74528 | .67965 ) 73353 || .69235 | 72156 | .70484|.70937| 11 
| 50 | .65336) . 75661 | 66697 |.74509 | .67987 73333 || .69256 | .72136 || .70505|.70916; 10 
51 | 65408! .75642 || .66718|.74489 | .68008) .73314 || 69277} .72116|| .70525|.70896) 9 
52 | 65430| 75623 | 66740 |.74470 | .68029] .73294 || .69298) .72095 |) .70546 | . 70875) 8 
53 | 65452! .75604 | .66762 74451 | .68051 | .73274 |) .69319 72075 || 70567} .70855| 7 
B41 | 65474! 75585 | 66783 . 74431 | .68072|.73254|| .69340|.72055 |) .70587'. 70884) 6 
55 | 65496| .75566|| .66805 .74412 | .68093 | .73234]| .69361 | .72035 || .70608'.70813) 5 
56 | 65518! 75547 || 60827 .74392 | 68115) .73215|| .69382| .72015 || .70628) 70793) 4 
57 | 65540. .75528 | .66848 .74373 | .68136|.73195|| .69403 | .71995 || . 70649 “70772| 8 
58 | 65562) .75509 || .66870 .74352 || .68157).73175 || .69424|.71974 || .70670 | 70752) 2 
59 | 65584) .75490 | .66891 .74334 | 68179 .73155 || .69445 | .71954 70690 .70731| 1 
60 | .65606 | .75471 |) .66913 74314 | 68200). 73135 || .69466 | .71934 70711|.70711| 0 
, (Cosin| Sine ||Cosin Sine | Cosin| Sine | Cosin | Sine | Cosin | Sine 
4g°.- || 47 46° {|= 45° 


49° | 


457 


TABLE XXVII.—NATURAL TANGENTS AND COTANGENTS. 


| el 0° Pe OO A ae 3° by 
Tang | Cotang Tang | Cotang || Tang | Cotang || Tang | Cotang | 
0; .00000 | Infinite.||} .0174 57.2500 || .03492 | 28.6363 |; .05241 | 19.0811 ‘60 
i} .00029 | 8437.75 || .01775 | 56.3506 |} .03521 | 28.3994 || .05270 | 18.9755 59 
2} .00058 | 1718.87 || .01804 | 55.4415 || .03550 | 28.1664 || .05299 | 18.8711 158 
8} .00087 | 1145.92 .01833 | 54.5613 || .03579 937. .05328 | 18.7678 157 
4} .00116 | 859.436 || .01862 | 53.7086 || .03609 | 27.7117 || .05857 | 18.6656 (56 
5} .00145 87.549 || .01891 | 52.8821 -03638 | 27.4899 .05387 | 18.5645 | 55 | 
6} .00175 | 572.957 || .01920 | 52.0807 || .08667 | 27.2715 || .05416 | 18.4645 154 | 
Z| 00204 | 491.106 || .01949 | 51.3082 || .03696 | 27.0566 |! .05445 | 18.3655 | 53 
8} .00233 | 429.718 || .01978 | 50.5485 || .03725 | 26.8450 |} .05474 | 18.2677 | 52 | 
9} .00262 | 381.97 .02007 | 49.8157 03754 | 26.6267 || .05503 | 18.1708 |5 
10} .00291 | 343.77 .02036 | 49.1039 || .03783 | 26.4316 || .05533 | 18.0750 50 
11; .CO0320 ) 312.521 || .02066 | 48.4121 || .03812 | 26.2296 || .05562 |. 17.9802 | 49 | 
12; .00349 | 286.478 || .02095 | 47.7395 || .03842 | 26.0807 || .05591 | 17.8863 | 48 | 
13} .00378 | 264.441 || .02124 | 47.0853 | .08871 | 25.8348 || .05620 | 17.7934 | 47 | 
14} .00407 | 245.552 || .02153 | 46.4489 |} .08900 | 25.6418 || .05649 | 17.7015 4G | 
15; .00486 | 229.182 02182 | 45.8294 |; .03929 | 25.4517 .05678 | 17.6106 | 45 | 
16| .00465 | 214.858 || .02211 | 45.2261 || .03958 | 25.2644 || .05708 | 17.5205 | 44 
Z| -00495 | 202.219 || .02240 | 44.6386 || .03987 | 25.0798 || .05737 | 17.4314 (43 
18} .00524 | 190.984 || .02269 | 44.0661 || .04016 | 24.897 .05766 | 17.8432 | 42 
19/ ,00553 | 180.982 || .02298 | 43.5081 || .04046 | 24.7185 || .05795 | 17.2558 |4 
20} .00582 | 171.885 || .02328 | 42.9641 || .04075 | 24.5418 || .05824 | 17.1693 | 40 
21} .00611 | 163.700 || .02357 | 42.4885 || .04104 | 24.3675 || .05854 | 17.0887 |39 
22} .00640 | 156.259 || .02386 | 41.9158 || .04138 | 24.1957 || .05883 | 16.9990 | 3: 
23} .00669 | 149.465 || .02415 | 41.4106 || .04162 | 24.0268 || .05912 | 16.9150 |37 
24] .00698 | 143.237 |! .02444 | 40.9174 || .04191 | 23.8593 || .05941 | 16.8319 136 
25| .00727 | 187.507 || .02473 | 40.4358 || .04220 | 23.6945 || .05970 | 16.7496 | 35 
26| .00756 | 182.219 || .02502 | 39.9655 || .04250 | 23.5321 |} .05999 , 16.6681 | 34 
27| .00785 | 127.32 02531 | 39.5059 || .04279 | 23.3718 || .06029 | 16.5874 |33 
28; .00815 | 122.77 -02560 | 39.0568 || .04308 | 23.2137 || .06058 | 16.5075 | 32 
29} .00844 | 118.540 || .02589 | 88.6177 || .04337 | 23.0577 || .06087 | 16.4283 | 31 
30; .00873 | 114.589 || .02619 | 38.1885 |) .04366 | 22.9038 || .06116 | 16.3499 |30 
31} .00902 | 110.892 || .02648 | 37.7686 || .04395 | 22.7519 I! .06145 | 16.2792 |2 
32] .00931 | 107.426 |} .02677 | 37.3579 N .04424 | 22.6020 || .06175 | 16.1952 | 9s 
33] .00960 | 104.171 |} .02706 | 86.9560 || .04454 | 22.4541 || .06204 ; 16.1190 | 27 
34). .00989 | 101.107 |; .02735 | 36.5627 || .04483 | 22.3081 || .06283 | 16.0435 |26 
85} .01018 | 98.217 02764 | 36.177 04512 | 22.1640 || .06262 | 15.9687 |25 | 
36; .01047 | 95.4895 || .02793 | 35.8006 || .04541 | 22.0217 || .06291 | 15.8945 |9 
37| .01076 | 92.9085 || .02822 | 35.4313 || .04570 | 21.8818 || .06321 | 15.8911 |93 
88! .01105 | 90.4633 .02851 | 35.0695 -04599 | 21.7426 -06350 | 15.7483 |22 
89) .01185 | 88.1436 -O2881 | 34.7151 .04628 | 21.6056 .06379 | 15.6762 | 21 
40} .01164 | 85.9398 || .02910 | 34.3678 || .04658 | 21.4704 || .06408 | 15.6048 | 20 
41} .01193 | 83.8435 || .02939 | 34.0273 || .04687 | 21.3369 || .06437 | 15.5340 | 19 
42} .01222,| 81.8470 || .02968 | 33.6935 || .04716 | 21.2049 || .06467 | 15.4638 118 
3| .01251 | 79.9434 .02597 | 33.3662 .04745 | 21.0747 .06496 | 15.3943 |17 
44} .01280 | 78.1265 .03026 | 33.0452 .04774 | 20 9460 .06525 | 15.3254 | 16 
45} .01309 | 76.3900 || .03055 | 32.7308 || .04803 | 20.8188 || .06554 | 15.2571 | 15 
46) .C1338 | 74.7292 || .03084 | 82.4213 || .04833 | 20.6932 || .06584 | 15.1893 114 
47} .01867 | 738.1390 .03114 | 32.1181 .04862 | 20.5691 .06613 | 15.1222 |13 
48) .01396 | 71.6151 || .03143 | 31.8205 || .04891 | 20.4465 || .06642 | 15.0557 112 
49; .01425 | 70.1533 .08172 | 31.5284 .04920 | 20.3253 .06671 | 14.9898 | 11 
50; .01455 | 68.7501 .08201 | 381.2416 .04949 | 20.2056 .06700 | 14.9244 |10 
‘1; .01484 | 67.4019 || .03230 | 30.9599 || .04978 | 20.0872 || .06730 | 14.8596:| 9 
2) .01513 | 66.1055 |} .03259 | 30.6833 || .059°7 | 19.9702 || .06759 | 14.7954 | 8 
3| .01542 | 64.8580 || .03288 | 30.4116 .05037 | 19.8546 .06788 | 14.7317! 7 
d4; .01571 | 68.6567 .03317 | 30.1446 || .05066 | 19.7403 .06817 | 14.6685 | 6 
55 | .01600 | 62.4992 .03846 | 29.8823 || .05095 | 19.6273 .06847 | 14.6059 | 5 
00} 01629 |’ 61.3829 .03376 | 29.6245 || .05124 | 19.5156 .06876 | 14.5438 | 4 
57) .01658 | 60.3058 -03405 | 29.3711 || .05153 | 19.4051 .06905 | 14.4823 | 3 
58) .01687 | 59.2659 .03434 | 29.1220 || .05182 | 19.2959 .06934 | 14.4212 | 2 
59| .01716 | 58.2612 -03463 | 28.8771 || .05212 | 19.1879 .06963 | 14.3607 | 1 
60} .01746 | 57.2900 .03492 | 28.6363 |} .05241 | 19.0811 | .06993 : 14.3007 | 0 
, Cotang) Tang ||Cotang| Tang ||Cotan g| Tang ||Cotang| Tang ; 
89° 88° 87° 86° 


458 


$$! 


PABLE XXVIUIL—NATURAL TANGENTS AND COTANGENTS. 


4° 


5° 


9, .07256 
10); .07285 


11| .07314 
12) .07344 


3| 07387: 
| 07402 
15| .074381 
16\ .07461 


—_ 
ie 


18| 07519 


t 

: 

: 

20| .0757 
| 21| .07607 
; 22| 07686 
i | 28] 07665 
07695 
251 07724 
26| .07753 
o7| 07782 
28| .07812 
| 29| .07841 
| 
: 


31) .07899 
32| .07929 
33| .07958 
34| .07987 
35| .08017 
26| .08046 
37| .08075 
88 |» .08104 


| .08192 
42 08221 
43| .08251 


ee 
re 


7| .07490 | 
19| .07548 | 


30| 07870 | 


a9| .08184 
40| 08163 


| 
| 


QQ) .06993 
£; .O07022 | 
2° 07051 
3 .07080 | 
4° ‘07110 | 
5) .07139 
6. .07168 
G 07197 | 
8! 07227 


| | 44| .08280 


45| .08309 


| 46| .98339 
47| .08368 | 


48| .08397 
49| .08427 

50| .08456 

+51! .08485 

52) .08514 
3 08544 


) 

/ 54} .08573 

155} .08602 
56! .08632 | 
| 57! .08661 

58} .08690 | 

159) .08720. | 

60) .08749 


| 


|Cotang 
/ 


14 


feed Red ee Re et Rt 


ras) 


vi) 


feck ek fk peek tek eh pk pk fk beh et fk ed fet pt bet 


11 


11 


~ Tang 


WMWNWWWWHW KCWWNWWWWW 


Ww wwe 


m0 We 


3007 
.2411 
.1821 
.1235 
.0655 
0079 


29507 | 
3.8940 | 


led 
.597 


3.7821 


T267 
6719 


6174 | 


56384 
.5098 
.4566 
.4039 
8515 


3.2996 


2480 
1969 
1461 
0958 
0458 
9962 
469 
8981 
8496 
8014 
7536 
7062 
6591 
6124 
£660 
5199 
4742 
4288 


.8390 
2946 
2905 


. 1632 
1201 
O72 
.0346 
9923 


9504 | 


9087 
8673 
8262 
7853 
7448 
. 7045 
6645 
6248 
.5853 


461 |} 


5072 | 


.4685 
.4301 


8858 | 


2067 || 


| 
le 
| Tang | Cotang 


Tang 


| .09746 


| 


Cotang | 


.10011 | 9 
.10040 | 9 
.10069 | 9 


.10099 | 9 
.10128 | 9 


.10510 | 9 
| Cotang 


Tang 


"08749 | 11.4301 
08778 | 11.3919 


| .08807 | 11.3540 
.08837 | 11.8168 
.O8866 | 11.2789 


| 08895 | 11.2417 
.08925 | 11.2048 
.08954 | 11.1681 
.08983 | 11.1816 
.09013 | 11.0954 
.09042 | 11.0594 


09071 | 11.0237 
"09101 | 10.9882 
| “09130 | 10.9529 
| "99159 | 10.9178 | 
| 09189 | 10.8829 | 
"09218 | 10.8483 
| "09247 | 10.8139 
| “09277 | 10.7797 
"09806 | 10.7457 
"09385 | 10.7119 


.09365 | 10.6788 
109394 | 10.6450 
(09423 | 10.6118 | 
'09453 | 10.5789 
09482 | 10.5462 
(09511 | 10.5186 
(09541 | 10.4813 
| :09570 | 10.4491 
09600 | 10.4172 
109629 | 10.3854 


| .09658 | 10.3538 
.09688 | 10.8224 
.09717 | 10.2913 | 
10.2602 
.09776 | 10.2294 
.09805 | 10.1988 | 
.09834 | 10.1683 
.09864 | 10.1881 
| .09893 | 10.1080 || 
.09923 | 10.0780 


.09952 | 10.0483 
.09981 | 10.0187 | 


.98981 
.96007 
.93101 
90211 
87338 


.51456 


.10158 | 9.84482 
| .10187 | 9.81641 
|| .10216 | 9.<8817 
10246 | 8.76009 | 
| 10275 | 9.73217 
| 10805 | 9.70441 
| .10334 | 9.67680 
 .10863 | 9.64935 || 
| 10393 | 9.62205 
10422 | 9.59490 
10452 | 9.56791 
10481 | 9.54106 


7e i \ 


| 85° 


|| 


84° 


459 


% SESAME cs PSO: PF 
Tang : Cotang | Tang: ! Cotang 
10510 | 9.51436 || .12278 | 8.14435 | 60 
10540 | 9.48781 || .12803 | 8.12451 |59 
10569 | 9.46144 || .12358 | 8.10586 | 58 
10599 | 9.43515 || .12867 | 8.08600 |57 
10628 | 9.40904 || .12507 | 8.06674 |56 

| .10657 | 9.88807 || .12426 | 8.04756 [55 

| 10687 | 9.85724 || .12456 | 8.02848 |54 
(10716 | 9.88155 || .12485 | 8.00948 |53 
10746 | 9.30599 || .12515 | 7.99058 |52 
10775 | 9.28058 || .12544 | 7.97176 |51 
10805 | 9.25530 || .12574 | 7.95802 |5 

| .10834 | 9.23016 || .12603 | 7.93488 | 49 

| 10863 | 9.20516 || .12683 | 7.91582 | 48 

| 10863 | 9.18028 || .12C62 | 7.89734 |47 
10922 | 9.15554 || .12692 | 7.87895 |46 
10952 | 9.18098 || .12722 | 7.86064 | 45 
10981 | 9.10646 || .12751 | 7.84242 | 44 

|| .11011 | 9.08211 || .12781 | 7.82428 | 43 

| (11040 | 9.05789 || .12810 | 7.80622 |42 
11070 | 9.08879 || .12840. | 7.78825 |41 
11089 | 9.00983 |} .12869 | 7.77035 | 40 

| .11128 | 8.98598 || .12899 | 7.75254 |39 
| (11158 | 8.96227 || .12929 | 7.73480 |38 
| .11187 | 8.93867 || .12058 | 7.71715 |387 
11217 | 8.91520 || .12988 | 7.69957 | 36 

| 11246 | 8.89185 || .18017 | 7.68208 | 35 

|| .112%6 | 8.86862 || .13047 | 7.66466 [34 
(11305 | 8.84551 || .12076 | 7.64782 | 33 

| 11885 | 8.82252 || .18106 | 7.63005 | 32 

| 112864 | 8.79964 || .18186 | 7.61287 |31 

| 11394 | 8.77689 || .18165 | 7.59575 | 30 

| .11423 | 8.75425 || .18195 7 57872. |29 

| .11452 | 8.73172 || .18224 | 7.56176 | 28 

| 11482 | 8.70981 || .18254 | 7.54487 [27 

| 11511 | 8.68701 || .18284 | 7.52806 | 26 

| 11541 | 8.66482 || .18813 | 7.511382 |25 

| 111570 | 8.64275 || 18843 | 7.49465 |24 
“11600 | 8.62078 || .18872° | 7.47806 | 23 
“11629 | 8.59893 || .13402 | 7.46154 | 22 
“11 1) | 8.57718 || .13482 | 7.44509 | 21 

| 114688 | 8.55555 || .13461 | 7.42871 | 20 

| 11718 | ¢ 58402 || .18491 | 7.41240 | 19 

| “11747 | 8.51259 || .13521 | 7.39616 |18 

| 114777 | 8.49128 |} .18550 | 7.387999 | 1% 
"11806 | 8.47007 || .18580 | 7.86889 | 16 

| -41886 | 8.44896 |} .13609 | 7.34786 | 15 

| “41865 | 8.42795 || .18639 | 7.33190 | 14 
"11895 | 8.40705 || .18669 | 7.31600 | 13 

| “41924 | 8.38625 || .18698 | 7.30018 |12 
"41954 | 8.36555 || .13728 | 7.28442 | 11 
"11983 | 8.34496 || .13758 | 7.26873 | 10 
12013 | 8.32446 |) .18787 | 7.25310 | 9 
"42042 | 8.30406 || .18817 | 7.23754 | 8 
"12072 | 8.28376 || .18846 | 7.22204 | 7 

| '12101 | 8.26355 || .18876 | 7.20661 | 6 
"42131 | 8.24345 || .18906 | 7.19125 | 5 

|| 12160 | 8.22844 || .18985 | 7.17594 | 4 
"42190 | 8.20852 || .18965 | 7.16071 | 3 

| 142219 | 8.18370 || .13995 | 7.14553 | 2 
“42249 | 8.16398 |, .14024 | 7.15042 | 1 

| 142978 | 8.14435 |! .14054 | 7.11537 | 0 

\Cotang| Tang | Cotang| Tang | | 

83° §2° 


~~ 


CHOIR OUMwWoHS| 


10 


_Tang 


14054 


- 14084 
14113 
. 14143 
-14173 
- 14202 
- 14232 
- 14262 
14291 
14321 
14351 
14381 
.14410 
- 14440) 
- 14470 
- 14499 
14529 
- 14559 
- 14588 
14618 
. 14648 
. 14678 
14707 
-14737 
14767 
14796 
14826 
. 14856 
- 14886 
- 14915 
- 14945 


.14975 
. 15005 
. 15034 
. 15064 
. 15094 
. 15124 
.15153 
. 15183 
. 15213 
. 15243 
115272 
. 15302 
. 15332 
. 15362 
. 15391 
. 15421 
. 15451 
. 15481 
15511 
. 15540 


. 15570 
. 15600 
. 15630 
. 15660 
15689 
.15719 
15749 
15779 


| .15809 


. 15838 


8° 


TABLE XXVIII—NATURAL TANGENTS AND COTANGENTS, 


9° 10° 11° 
Cotang |} Tang | Cotang || Tang Cotang || Tang | Cotang 
7.11587 || .15888 | 6.31375 .17633 | 5.67128 -19438 | 5.14455 
7.10038 .15868 | 6.30189 .17663 | 5.66165 -19468 | 5.13658 
7.08546 -15893 | 6.29007 |} .17693 | 5.65205 .19498 | 5.12862 
7.07059 . 15928 . 27829 17723 | 5.64248 -19529 | 5.12069 
7.05579 -15958 | 6.26655 || .17753 | 5.63295 -19559 | 5.11279 
7.04105 -15988 | 6.25486 17783 .62544 || .19589 | 5. 10499 
7.02637 -16017 | 6.24321 .17813 | 5.613897 -19619 | 5.09704 
7.01174 16047 | 6.23160 || .17843 | 5.60452 -19649 | 5.08921 
6.99718 |; .16077 | 6.22003 || .17873 | 5.59511 -19680 | 5.08139 
6.98268 .16107 | 6.20851 .17903 | 5.58573 :19710 | 5.07360 
6.96823 |} .16137 | 6.19703 |] .17933 | 5.57638 |! .19740 5.06584 
6.95385 || .16167 | 6.18559 || .17963 5.56706 || .19770 | 5.05809 
6.93952 |} .16196 | 6.17419 || .17998 | 5.5577 -19801 | 5.05037 
6.92525 || .16226 | 6.16283 || .18023 | 5.54851 |] .19881 5.04267 
6.91104 || .16256 | 6.15151 |} .18053 | 5.53927 || .19861 5.03499 
6.89688 |/ .16286 | 6.14023 || .18083 | 5.53007 || .19891 5.02784 
6.88278 - 16316 | 6.12899 .18113 | 5.52090 -19921 | 5.01971 
6.86874 || .16346 | 6.1177 -18143 | 5.51176 || .19952 | 5.01210 
6.85475 -16376 | 6.10664 || .18173 | 5.50264 || .19982 5.00451 
6.84082 || .16405 | 6.09552 || 18208 | 5.49356 -20012 | 4.99695 
6.82694 || .16435 | 6.08444 || .18233 | 5.48451 || 20042 4.98940 
6.81312 || .16465 | 6.07340 || .18263 | 5.47548 || .20073 4.98188 
6.79936 || .16495 | 6.06240 ;| .18293 .46648 -20103 | 4.97438 
6.78564 || .16525 | 6.05143 || .18323 | 5.45751 -20133 | 4.96690 
6.77199 || .16555 | 6.04051 || .18353 | 5.44857 -20164 | 4.95945 
6.75838 || .16585 | 6.02962 |} .18384 | 5.43966 -20194 | 4.95201 
6.74483 || .16615 | 6.01878 |] .18414 | 5.48077 || _20224 4.94460 
6.73133 || .16645 | 6.00797 || .18444 | 5.42192 -20254 | 4.93721 
6.71789 || .16674 | 5.99720 || .18474 | 5.41309 || .20985 4.92984 
6.70450 || .16704 | 5.98646 || .18504 | 5.40429 -20315 | 4.92249 
6.69116 16734 | 5.97576 || .18534 | 5.39552 || .20345 4.91516 
6.67787 || .16764 | 5.96510 |! .18564 | 5.38677 || 20376 4.90785 
6.66463 || .16794 | 5.95448 || .18594 | 5.37805 || .20406 4.90056 
6.65144 |/ .16824 | 5.94390 |} .18624 | 5.36936 || .20436 4.89330 
6.63831 || .16854 | 5.93385 || .18654 | 5.3607 -20466 | 4.88605 
6.62523 || .16884 | 5.92283 || .18684 | 5.35206 || .20497 4.87882 
6.61219 .16914 | 5.91236 18714 | 5.34345 -20527 | 4.87162 
6.59921 || .16944 | 5.90191 || .18745 | 5.33487 || .20557 | 4 86444 
6.58627 || .16974 | 5.89151 18775 | 5.32631 .20588 | 4.85727 
6.57339 .17004 | 5.88114 18805 | 5.31778 || .20618 | 4.85013 
6.56055 |} .17033 | 5.87080 || .18835 | 5.30928 |! .20648 4.84300 
6.54777 || .17063 | 5.86051 || .18865 | 5.30080 |] .20679 4.83590 
6.53503 || .17093 | 5.85024 -18895 | 5.29235 -20709 | 4.82882 
6.52234 || £17123 | 5.84001 18925 | 5.28393 || .20739 | 4.82175 
6.50970 || .17153 | 5.82982 || .18955 | 5.27553 || /20770 | 4.81471 
6.49710 17183 | 5.81966 || .18986 | 5: 26715 -20800 || 4.80769 
6.48456 17213 | 5.80953 || .19016 | 5.25880 -20830 | 4.80068 
6.47206 17243 | 5.79944 -19046 | 5.25048 || .20861 | 4.79370 
6.45961 17278 | 5.78938 -19076 | 5.24218 -20891 4.78673 
6.44720 || .17303 | 5.77936 -19106 | 5.23391 |' .20921 | 4.77978 
6.43484 . 17333 | 5.76937 -19136 | 5.22566 || .20952 | 4. 77286 
6.42253 || .17363 | 5.75941 || .19166 | 5.21744 || .20982 | 4.76595 
6.41026 || .17893 | 5.74949 .19197. | 5.20925 -21013 | 4.75906 
6.39804 |) .17423 | 5.73960 || .19227 | 5.20107 .21043 | 4.75219 
6.38587 || .17453 | 5.72074 -19257 | 5.19293 -21073 | 4.74534 
6.3737 .17483 | 5.71992 .19287 | 5.18480 .21104 | 4.73851 
6.36165 7513 | 5.71013 229817): Del T6%1 21134 | 4.73171 
6.34961 .17548 | 5.70037 .19347 | 5.16863 .21164 | 4.72490 
6.33761 || .17573 | 5.69064 1987 5.16058 -21195 | 4.71813 
6.32566 || .17603 | 5.68094 || .19408 | 5.15256 || .212295 | 4.71137 
6.31375 . 17683 | 5.67128 .19438 | 5.14455 || .21256 | 4.70463 
\Cotang| Tang , Cotang| Tang ||Cotang| Tang Cotang| Tang 
81° | 80° 79° 78° 


eeeaeniaae = 


TABLE XXVIIIL—NATURAL TANGENTS AND COTANGENTS. 


| ry 19° 13° jae | 15° 
: Tang Cotang || Tang | Cotang |; Tang | Cotang || Tang | Cotang 
| Q| .21256 | 4.70463 || .23087 | 4.33148 || .24933 | 4.01078 || .26795 | 3.73205 
| 1] .21286 | 4.69791 || .23117 | 4.82573 || .24964 | 4.00582 || .26826 | 3.7277 
| 2) .21816 | 4.69121 .28148 | 4.32001 .24995 | 4.00086 || .26857 | 3.42338 
3} .213847 | 4.68452 || .23179 | 4.31430 || .25026 | 3.99592 || .26888 | 3.71907 
4} .21877 | 4.67786 || .23209 | 4.80860 || .25056 | 3.99099 || .26920 | 8.71476 | 
5} .21408 | 4.67121 .23240 | 4.30291 .25087 | 3.98607 || .26951 | 3.71046 
6| .21438 | 4.66458 || .28271 |. 4.29724 || .25118 | 3.98117 || .26982 | 3.70616 
7) .21469 | 4.65797 |} .283801 | 4.29159 || .25149 | 3.97627 || .27013 | 3.70188 
8} .21499 | 4.65188 || .2383832 | 4.28595 || .25180 | 3.97189 || .27044 | 3.69761 
9} .215¢9 | 4.64480 || .23363 | 4.28032 || .25211 | 8.96651 .27076 | 3.69385 
10} .21560 | 4.63825 || .28393 | 4.27471 || .25242 | 3.96165 || .27107 | 3.68909 
11} .21590 | 4.63171 .23424 | 4.26911 .25273 | 3.95680 || .27188 | 3.68485 
12} .21621 | 4.62518 || .23455 | 4.26852 || .25804 | 3.95196 ‘| .27169 | 3.68061 
| 13} .21651 | 4.61868 23485 | 4.25795 || .25385 | 3.94718 || .27201 | 3.67638 
14! .21682 | 4.61219 23516 | 4.25289 || .25366 | 3.94282 || .27282 | 3.67217 
15} .21712 | 4.60572 23547 | 4.24685 || .25897 | 3.98751 .27263 | 3.66796 
| 16] .21743 | 4.59927 23578 | 4.24182 || .25428 | 3.93271 .27294 | 3.66376 
17| .21773 | 4.59283 23608 | 4.28580 || .25459 | 3.92798 || .273826 | 3.65957 
18) .21804 | 4.58641 23639 | 4.23080 || .25490 | 3.92816 || .27357 | 8.65538 
19} .21834 | 4.58001 23670 | 4.22481 .25521 | 3.91839 || .27888 | 8.65121 
bal -21864 | 4.57363 23700 | 4.21983 || .25552 | 3.91364 |} .27419 | 8.64705 
21) .21895 | 4.56726 28731 | 4.21887 || .25583 | 8.90890 || .27451 | 8.64289 
22} .21925 | 4.56091 23762 | 4.20842 || 25614 | 3.90417 || .27482 | 3.63874 
23) .21956 | 4.55458 23793 | 4.20298 || .25645 | 3.89945 || .27513 | 3.63461 
| 24] .21986 | 4.54826 238823 | 4.19756 || .25676 | 8.89474 || .27545 | 3.63048 
25} -.22017 | 4.54196 23854 | 4.19215 || .25707 | 3.89004 || .27576 | 3.62636 
26) .22047 | 4.53568 28885 | 4.18675 || .25738 | 3.885386 || .27607 | 8.62224 
Q7| .22078 | 4.52941 23916 | 4.18187 || .25769 | 8.88068 || .27638 | 3.61814 
28} .22108 | 4.52316 23946 | 4.17600 ||} .25800 | 3.87601 .27670 | 3.61405 
29| .22139 | 4.51693 23977 | 4.17064 || .258381 | 3.87136 || .27701 | 8.60996 
| 30| .22169 | 4.51071 24008 | 4.16530 || .25862 | 3.86671 || .277382 | 3 60588 | 
| 81} .22200 | 4.50451 24039 | 4.15997 |} .258938 | 3.86208 || .27764 | 3.60181 
382| .22231 | 4.49832 24069 | 4.15465 |} .25924 | 3.85745 || .27795 | 8.59775 
33] .22261 | 4.49215 24100 | 4.14984 || .25955 | 8.85284 || .27826 | 3.59370 
| 34| .22292 | 4.48600 || .24181 |°4.14405 || .25986 | 3.84824 || .27858 | 3.58966 
35] .22322 | 4.47986 24162 | 4.18877 || .26017 | 3.84364 |} .27889 | 3.58562 
386} .22353 | 4.47374 || .24193 | 4.13850 || .26048, | 3.83906 || .27921 | 8.58160 
37| .223883 | 4.46764 || .24223 | 4.12825 26079 | 3.83449 || .27952 | 3.57758 
38] 22414 | 4.46155 24954 | 4.12301 .26110 | 3.82992 219838 | 3.57357 
39| .22444 | 4.45548 24285 | 4.11778 || .26141 | 8.82587 || .28015 | 3.56957 
40| .22475 | 4.44942 24316 | 4.11256 || .26172 | 3.82088 || .28046 | 3.56557 
41} .22505 | 4.44338 || .243847 | 4.10786 || .26203 | 3.81630 || .28077 | 3.56159 
. 42) 1.22536 | 4.48785 || .243877 | 4.10216 || .26235 | 3.81177 || .28109 | 3.55761 
} 43] .22567 | 4.43134 |) .24408 | 4.09699 || .26266 | 3.80726 || .28140 | 3.55364 
| 44| .22597 | 4.42534 .24489 | 4.09182 .26297 | 3.80276 28172 | 3.54968 
45| 122628 | 4.41936 || .24470 | 4.08666 || .26328 | 8.79827 || .28203 | 8.54573 
. 46, .22658 | 4.41340 || .24501 | 4.08152 || .26359 | 3.79378 || .28234 | 3.54179 
| 47| .22689 | 4.40745 245382 | 4.07639 |} .26890 | 3.778931 .28266 | 3.53785 
i 48! .22719 | 4.40152 24562 | 4.07127 || .26421 | 8.78485 || .28297 | 3.53393 
49; .22750 | 4.39560 24593 | 4.06616 || .26452 | 3.78040 || .28329 | 3.53001 
| 50) .22781 | 4.388969 24624 | 4.06107 || .26483 | 3.77595 || .28860 | 3.52609 
51) .22811 | 4.38381 24655 | 4.05599 |} .26515 | 3.77152 || .283891 | 3.52219 
52) .22842 | 4.37793 24686 | 4.05092 || .26546 | 3.76709 || .28423 | 3.51829 
53] .22872 | 4.37207 24717 | 4.04586 |} .26577 | 3.76268 || .28454 | 3.51441 
54! .22903 | 4.36623 || .24747 | 4.04081 || .26608 | 3.75828 || .28486 | 3.51053 
55} .22924 | 4.36040 || .24778 | 4.038578 || .26689 | 8.753888 || .28517 | 3.50666 
56] .22964 | 4.35459 24809 | 4.03076 |} .26670 | 3.774950 || .28549 | 3.5027’ 
57|_.22995 | 4.34879 24840 | 4.0257. .26701 | 3.74512 || .28580 | 3.49894 
58| .23026 | 4.34300 24871 | 4.02074 || .26783 | 3.74075 || .28612 | 3.49509 
59} .28056 | 4.33723 24902 | 4.01576 || .26764 | 3.73640 || .28643 | 8.49125 
60) .238087 | 4.33148 24933 | 4.01078 || .26795 | 3.738205 ||} .28675 | 3.48741 
\Cotang| Tang | Cotang Tang ||Cotang| Tang |/Cotang| Tang 
/ ee ee eee ee ee iat ae tne bead EE hes ' 
| Pea - | 76° | 75° 74° 


lomeocwmoa20w 


‘ 16° 17° 18° \| 19° : 
Tang | Cotang || Tang | Cotang |; Tang | Cotang || Tang | Cotang 
QO} .28675 | 3.48741 || .80573 | 5.27085 .82492 | 3.07768 84433 | 2.90421 |60 | 
1} .28706 | 3.48359 || .80605 | 3.26745 || .82524 | 3.07464 || .384465 | 2.90147 |59 
2| .28788 | 3.47977 || .380687 | 3.26406 || .82556 | 3.07160 || .84498 | 2.89873 |58 
8| .28769 | 3.47596 || .80669 | 3.26067 || .382588 | 3.06857 || .384530 | 2.89600 |57 
4; .28800 | 3.47216 || .380700 | 3.25729 || .82621 | 3.06554 || .384563 | 2.89827 |56 
5| .288382 ) 3.46837 || .80782 | 3.25892 || .82653 | 3.06252 || .84596 | 2.89055 155 
6| .28864 | 8.46458 || .80764 | 3.25055 || .82685 | 3.05950 || .384628 | 2.88788 |54 
7| .28895 | 3.46086 || .380796 | 3.24719 || .382717 | 3.05649 || .34661 | 2.88511 | 53 
8} .28927 | 8.45703 || .80828 | 3.24383 || .32749 | 3.05349 || .34693 | 2.88240 | 52 
9} .28958 | 8.45327 || .80860 | 3.24049 || .82782 | 3.05049 || .34726 | 2.87970 |51 
10; .28990 | 3.44951 |} .80891 | 3.28714 || .82814 | 3.04749 || .84758 | 2 87700 |50 
11) .29021.| 3.44576 || .80923 | 3.23381: || .82846 | 3.04450 || .84791 | 2.87430 | 49 
12} .29053 | 3.44202 || .80955 | 3.23048 |, .82878 | 3.04152 |; .84824 | 2.87161 | 48 
13| .29084 | 3.48829 || .80987 | 3.22715 || .82911 | 3.03854 || .384856 | 2.86892 | 47 
14} .29116 | 3.43456 || .81019 | 3.22884 || .382943 | 3.038556 || .384889 | 2.86624 | 46 
15} .29147 | 3.43084 || .81051 | 3.22053 || .82975 | 3.08260 || .34922 | 2.86356 | 45 
16} .29179 | 3.42713 || .381083 | 3.21722 || .388007 | 3.02963 || .84954 | 2.86089 | 44 
7| .29210 | 3.42343 || .81115 | 3.21892 || .338040 | 3.02667 || .384987 | 2.85822 | 43 
18} .29242 ) 3.41973 || .81147 | 3.21068 || .388072 | 3.02872 || .385020 | 2.85555 | 42 
19} .29274 | 3.41604 || .81178 | 3.20734 || .33104 | 3.02077 || .35052 | 2.85289 | 41 
20} .29805 | 3.41286 |) .81210 | 3.20406 || .83186 | 3.01783 |; .85085 | 2.85023 | 40 


21) .29337 | 3.40869 || .81242 | 3.20079 || .33169 | 8.01489 || .385118 | 2.84758 | 39 
a 22| .29368 | 3.40502 || .81274 | 3.19752 || .388201 | 8.01196 || .85150 | 2.84494 | 38 
2 | 23| .29400 | 3.40136 || .81806 | 3.19426 || .383233 | 3.00903 || .85183 | 2.84229 | 37 
24| .29432 | 3.3977 .31338 | 3.19100 || .83266 | 3.00611 || .85216 | 2.83965 | 36 
Ha a8 25} .29463 | 3.39406 || .381870 | 3.18775 || .338298 | 3.00819 || .85248 | 2.83702 | 35 

26| .29495 | 8.39042 || .381402 | 3.18451 || .383330 | 3.00028 || .85281 | 2.83439 | 34 
27| .29526 | 3.88679 || .81434 | 3.18127 || .383363 99738 || .385314 | 2.83176 | 33 


97144 || .85608 | 2.80833 | 24 

96858 || .85641 | 2.80574 | 23 | 
96573 || .85674 | 2.80316 | 22 | 
96288 || .85707 80059 | 21 | 
.96004 || .85740 


36/ .29811 
37| .29848 
38| .29875 


.00443 || .381722 
85087 || .81754 .14922 || .38686 


3 

3 

3 

3 

3 

3 

3 

3.15240 || .38654 

3 
84732 || .81786 | 3.14605 || .33718 

3 

3 

3 

3 

3 

3 

3 

3 

3 


i 28| .29558 | 3.38317 || .81466 | 3.17804 || .38395 99447 || .85346 | 2.82914 | 32 
Hy 29; .29590 | 3.387955 || .81498 .17481 || .83427 99158 || .85379 | 2.82653 | 31 
ap 80] .29621 | 3.387594 || .81530 .17159 .|| .33460 98868 || .385412 | 2.82391 | 30 
Hee ia i 31} .29653 | 3.37234 || .31562 .16838 || .383492 98580 || .85445 | 2.82130 | 29 
aie 82] .29685 | 3.36875 || .381594 .16517 || .33524 98292 || .85477 | 2.81870 | 28 
bo 33} .29716 | 3.36516 || .31626 .16197 || .388557 98004 || .85510 | 2.81610 | 27 
Tan eh |i 34| .29748 | 8.36158 || .31658 15877 || .3838589 9717 || .85543 | 2.81350 | 26 
He 85] .29780 : .85800 || .81690 .15558 |} .88621 97430 || .85576 | 2.81091 | 25 

3 

3 


ii 39} .29906 
He | 40} .29938 


WWWWNWNWWWWW WWWYW 


34377 || .81818 14288 |} .33751 
34023 || .81850 | 3.13972 || .83783 


% 0 


79802 120} 


3. 
3. 
a 41| .29970 | 3.33670 || .81882 | 3.18656 || .33816 | 2.95721 || .85772 | 2.79545 119 
aaah 2| .30001 | 3.33317 || .81914 | 3.13841 || .83848 | 2.95437 || .85805 | 2.79289 |18| | 
AAT 43| .30038 | 3.32965 || .81946 | 3.18027 || .88881 | 2.95155 || .85838 | 2.79033 17] 
au 44| .30065 | 8.32614 || .81978 | 3.12713 || .83913 | 2.94872 || .85871 | 2.78778 |16} _ 
Nae 45) .30097 | 3.32264 || .82010 | 3.12400 || .33945 | 2.94591 |] .85904 | 2.78523 /15] 
Mane | 46} .30128 | 8.31914 || .82042 | 3.12087 || .33978 | 2.94309 || .85937 | 2.78269 | 14 
| 47| .30160 | 8.31565 || .82074 | 3.11775 || .84010 | 2.94028 || .35969 | 2.78014 |13 
48) .30192 | 8.31216 || .82106 | 3.11464 || .34043 | 2.98748 || .36002 | 2.77761 |12 
ij 4)| .30224 | 3.30868 || .382189 | 8.11153 || .34075 | 2.93468 || .86035 | 2.77507 | 11 
50; .80255 | 8.30521 || .82171 | 8.10842 || .34108 | 2.93189 || .386068 | 2.77254 | 10 
51| .30287 | 3.30174 || .82208 | 3.10582 || .84140 | 2.92910 || .36101 | 2.77002-| 9 
52] .80319 | 8.29829 || .82235 | 3.10223 |} .34173 | 2.92632 || .36134 | 2.76750 | 8 
53] .80351 | 3.29488 || .82267 | 3.09914 |] .34205 | 2.92354 || .86167 | 2.76498 | 7 
54| .30382 | 3.29139 |} .82299 | 3.09606 || .34238 | 2.92076 || .36199 | 2.76247 | 6 
55| .80414 | 3.28795 || .32831 | 3.09298 || .84270 | 2.91799 || .36232 | 2.75996 | 5 
56| .30446 | 8.28452 || .82363 | 3.08991 || .34303 | 2.91523 || .86265 | 2.75746 | 4 
57| .80478 | 8.28109 || .82396 | 3.08685 || .34335 | 2.91246 || .36298 | 2.75496 | 3 
58] .30509 | 8.27767 || .32428 | 3.08379 || .34368 | 2.90971 || .36331 | 2.75246 | 2 
59| .80541 | 8.27426 || .82460 | 8.08073 || .34400 | 2.90696 || .36364 | 2.74997 | 1] | 
60} .80573 | 3.27085 | 32492 | 8.07768 |] .34433 | 2.90421 || 36397 | 2.74748 | 0] 
Cotang| Tang CObeaNS Tang |/Cotang| Tang || Cotang | Tang ; | 
de eee nn a ee eer | One Deere. Saeeee enna | PO ene Een eee ||| tene |e al ee 
73° | 72° | "1° | 70° 
ln re sn ee es ee 


462 


TABLE XXVII.—NATURAL TANGINTS AND COTANGINTS. 


| 


20° || ie 22° | 23° 


|! Tang | Cotang || Tang | Cotang!; Tang | Cotang 
60509 || .40408 | 2.47500 ;| .42447 | 2.85585 
60283 .40486 | 2.473802 || .42482 | 2.85395 
60057 || .40470 .47095 || .42516 | 2.85205 
593831 || .40504 | 2.46888 || .42551 .80015 
.88520 | 2.59606. || .40588 | 2.46682 || .42585 | 2.34825 
£88558 59381 .40572 | 2.46476 || .42619 | 2.34636 
.88587 .59156 || .40606 | 2.4627 .42654 84447 
.88620 58932 || .40640 | 2.46065 || .42688 . 84258 
88654 58708 || .40674 .4586 42722 | 2.34069 
88687 58484 || .40707 | 2.45655 42757 83881 
38721 .58261 || .40741 .4545 .42791 383693 
.88754 .58038 || .4077 452 .42826 33505 
.88787 57815 |; .40809 .45043 |} .42860 BBol? 
80821 57593 || .40843 .44839 || .42894 . 83180 
88854 573871 || .40877 «4 .42929 82943 
| .38888 57150 || .40911 444: -42963 82756 
.88921 56928 .40945 | 2.442% .42998 .82570 
88955 56707 || .40979 .44027 || .48082 82383 
88988 .56487 || .41013 .43825 || .48067 82197 
| .39022 56266 || .41047 365 .43101 .82012 
|| .89055 56046 || .41081 .43136 


.31826 
.69612 || .39089 55827 |} .41115 4B R% .43170 31641 
.69371 .89122 55608 |; .41149 43015 43205 .81456 
.69131 || .89156 55389 || .41183 428 43239 81271 
.68892 ||} .389190 5517 41217 43274 381086 
.68653 |; .38922: 54952 || .41251 .43308 80902 
.68414 || 89257 54734 41285 43343 80718 
.68175 || .39290 54516 || .41819 43378 80534 
O7T93T || .89324 54299 || .41853 .43412 80351 
.67709 || .89357 54082 || .41887 43447 80167 
.67462 || .89091 .58865 || .41421 43481, 


29984 
67225 || .39425 41455 41223 || .48516 .29801 | 
66989 || .389458 .41490 41025 || .48550 29619 
.66752 || .89492 41524 40827 || .48585 29437 12 
.66516 || .89526 41558 4062 .43620 29254 |% 
.66281 |! .89559 .41592 .40482 || .48054 29073 | 2 
.66046 |) .89593 .41626 .40285 || .438689 28891 
.65811 || .89626 .41660 .400388 |) 48724 28710 | 2 
.65576 || ~.89660 741604 89841 || .48758 28528 | 2% 
65342 || .389694 41728 .89645 || .48793 28348 
.65109 || .39727 41763 89449 || .43828 28167 
.64875 |} .389761 1797 89253 || .48862 27987 
64642 || .39795, 41831 39058 || .43897 27806 
64410 || .89829 41865 .88863 || .45982 27626 
64177 |; .389862 41899 .88668 || .48966 27447 
63945 || .89896 41933 .88473 || .44001 27267 
.63714 |! .89930 .41968 88279 || .44036 27088 
.63483 || .89963 .42002 | 2.38084 || .44071 | 2.26909 
63252 || .89997 .42036 87891 || .44105 26730 
63021 || .40031 49807 || .42070 | 2.87697 || .44140 26552 | 
62791 || .40065 49597 || .42105 .87504 || 44175 2637 
62561 || .40098 | 2.49886 || .42139 37311 || .44210 26196 
62332 || .40132 49177 || .421%73 87118 || .44244 | 2.26018 
62103 || .40166 A8967 || .42207 | 2.36925 |} .44279 2584 
.61874 || .40200 48758 || .42242 | 2.36783 || .44814 25663 
61646 || .40234 | 2.48549 || .42276 .86541 || .44849 25486 
61418 || .40267 | 2 48340 || .42310 36349 || .44884 25609 
.61190 || .40301 48132 || .42345 | 2.36158 || .44418 25132 
.60963 || .40335 | 2.47924 || .42379 85967 || .44453 24956 
60736. || .40369 | 2.47716 || .42413 85776 .44488 | 2.24780 
60509 || .40403 | 2.47509 || .42447 | 2.85585 || .44523 2.24604 
Tang ||Cotang; Tang | Cotang| Tang \Cotang | Tang 
) 68° 67° 
463 


~ 


83336 
|| 88420 | 
| .88453 
38487 


Co) 


W 


0 ®D WW WWW 


oomwmnc! 


36661 
36694 
86727 


.386760 
.86793 
. 36826 | 
. 86859 
80892 
| .86925 
.86958 
.386991 
| .387024 
20| .37057 
.387090 
.87123 
Sole 
.387190 
81223 
£31256 
631289 
eolooe | 
.o1 800 
£81388 
O1422 
331455 
.37488 
of b21 
| .87554 
36| .87588 
37621 
| 37654 
1687 
iS Ci20 
orto 
118" 
| .37820 
.31 803 
5| .387887 
| .37920 
| .87953 
3} 387986 
J} .38020 
50| .88053 
| .88086 
| .88120 
| .38153 
| .88186 
-38220 
5) .388253 
| .38286 
. 38320 
| .383853 
88386 


[We COUOH 


WOWNWNWNWNWNWNWWYWD ww 


9 W 0 


WWWNWWWWD 


9% WW W 09 0 WW WNWNWNWNWNWNWNWNWW NYONWNWNWNWWNWNWNWYD WNWWNWWWWi 
WWWWNWWWWNWYW WNWWNWNWNWNWNWWWD WNWNWNWNWNWNWNNWNWD WWWW 


ru) 4] WWWWNWWWNW WNWNWWWW 


D0 tO WW 


WWWWNWNDWNWWW*W 


WMWNWNWWNHW WW WW 


WNWNWNWNWNWNWNWNDW NWWWNWNWNWNVNWND WWW NWNVNWNWNWW 


VWNWNWWNWWWWO WWW WNWWNWNWWW 


WNWNWWNWNWWWWW 
lomrmcmornIwH0© 


~ 


TABLE XXVIIL—NATURAL TANGENTS AND COTANGENTS, 


; 24° | 25° 26° | 27° | P 
Tang | Cotang || Tang | Cotang |; Tang | Cotang || Tang | Cotang 
O| .44523 | 2.24604 .46631 | 2.14451 || .48778 | 2.05080 || .50953 | 1.96261 | 60} 
* J} .44558 | 2.24428 .46666 | 2.14288 .48809 | 2.04879 || .50989 | 1.96120 |59 | 
2] .44593 | 2.24252 46702 | 2.14125 || .48845 | 2.04728 || .51026 | 1.95979 [58 | 
3| .44627 | 2.24077 || .46787 | 2.13963 || .48881 | 2.04577 || .51063 | 1.95838 [57 | 
4| .44662 | -2.23902 .46772 | 2.138801 48917 | 2.04426 .51099 | 1.95698 | 56 | 
5| .44697 | 2.23727 || .46808 | 2.13639 || .48953 | 2.0427 .51186 | 1.95557 | 55 | 
6| .44782 | 2.23553 || .46843 | 2.13477 || .48989 | 2.04125 }| .51173 | 1.95417 | 54) 
7| 44767 | 2.23378 || .46879 | 2.18316 || .49026 | 2.03975 || .51209 | 1.95277 153 | 
8| .44802 | 2.23204 || .46914 | 2.18154 || .49062 | 2.03825 || .51246 | 1.951387 | 52 | 
9| .44837 | 2.23030 || .46950 | 2.12993 || .49098 | 2.03675 || .51288 | 1.94997 |51 | 
10| .44872 | 2.22857 .46985 | 2.12882 || .49184 | 2.03526 || .51319 | 1.94858 |50 | 
11| .44907 | 2.22683 |! .47021 | 2.12671 || .49170 | 2.03376 || .513856 | 1.94718 | 49 
2) .44942 | 2.22510 | .47056 | 2.12511 || .49206 | 2.08227 || .51393 | 1.94579 |48 
13) .44977 | 2.22337 || .47092 | 2.12350 || .49242 | 2.08078 || .51430 | 1.94440 | 47 | 
14| .45012 | 2.22164 || .47128 | 2.12190 || .49278 | 2.02929 || .51467 | 1.94301 |46 
15| .45047 | 2.21992 || .47163 | 2.12030 || .49315 | 2.02780 || .51503 | 1.94162 | 45 
16| .45082 | 2.21819 || .47199 | 2.11871 || .49351 | 2.02681 || .51540 | 1.94023 | 44 
17| .45117 | 2.21647 || .47234 | 2.11711 || .49387 | 2.02483 || .51577 | 1.98885 |43 
18) .45152 | 2.21475 || .47270 | 2.11552 || .49423 | 2.02385 || .51614 | 1.93746 | 42 
19| .45187 | 2.21304 || .47805 | 2.11892 || .49459 | 2.02187. || .51651 | 1.93608 | 41 
20| .45222 | 2.21182 || .47841 | 2.11288 || .49495 | 2.02039 || .51688 | 1.93470 | 40 
21| .45257 | 2.20961 || .473877 | 2.11075 || .49532 | 2.01891 || .51724 | 1.98332 |39 
92] .45292 | 2.20790 || .47412 | 2.10916 || .49568 | 2.01743 || .51761 | 1.93195 | 38 | 
93] .45327 | 2.20619 || .47448 | 2.10758 || .49604 | 2.01596 || .51798 | 1.93057 | 37 
24| .45362 | 2.20449 || .47483 | 2.10600 || .49640 | 2.01449 || .51835 | 1.92920 | 36 
25] .45397 | 2.20278 || .47519 | 2.10442 || .49677 | 2.01302 || .51872 | 1.92782 |35 
26| .45432 | 2.20108 || .47555 | 2.10284 || .49713 | 2.01155 || .51909 | 1.92645 | 34 
27| .45467 | 2.19938 || .47590 | 2.10126 || .49749 | 2.01008 || .51946 } 1.92508 | 33 
28} .45502 | 2.19769 .47626 | 2.09969 || .49786 | 2.00862 |} .51983 | 1.92871 |32 
29| .45538 | 2.19599 47062 | 2.09811 || .49822 | 2.00715 .52020 | 1.922385 131 
30| .45573 | 2.19430 || .47698 | 2.09654 || .49858 | 2.00569 |} .52057 | 1.92098 | 30 
31| .45608 | 2.19261 || .47783 | 2.09498 || .49894 | 2.00423 || .52094.| 1.91962 |29 
32] .45643 | 2.19092 || .47769 | 2.09841 || .49931 | 2.00277 || .52131 | 1.91826 |28 
83| .45678 | 2.18928 47805 | 2.09154 .49967 | 2.00131 .52168 | 1.91690 |27 
84] .45713 | 2.18755 || .47840 | 2.09028 || .50004 | 1.99986 |} .52205 | 1.91554 | 26 
85! .45748 | 2.18587 || .47876 | 2.08872 || .50040 | 1.99841 || .52242 | 1.91418 | 25 
36| .45784 | 2.18419 || .47912 | 2.08716 || .50076 | 1.99695 || .52279 | 1.91282 | 24 
| .45819 | 2.18251 .47948 | 2.08560 |} .50118 | 1.99550 52316 | 1.91147 | 2¢ 
88) .45854 | 2.18084 .47984 | 2.08405 || .50149 | 1.99406 52353 | 1.91012 | 2; 
39] .45889 | 2.17916 .48019 | 2.08250 .50185 | 1.99261 .52390 | 1.90876 | 21 
40| .45924 | 2.17749 || .48055 | 2.08094 |} .50222 | 1.99116 || .52427 | 1.90741 |2 
41| .45960 | 2.17582 || .48091 | 2.07989 || .50258 | 1.98972 || .52464 | 1.90607 | 19 
42| .45995 | 2.17416 || .48127 | 2.07785 || .50295 | 1.98828 || .52501 | 1.90472 |18 
43| .46030 | 2.17249 || .48163 | 2.07630 || .50331 | 1.98684 || .52538 | 1.90337 |17 
44| .46065 | 2.17083 || .48198 | 2.07476 || .50368 | 1.98540 || .52575 | 1.90203 | 16 | 
45| .46101 | 2.16917 || .48234 | 2.07821 || .50404 | 1.98396 || .52613 | 1.90069 |15 
46| .46186 | 2.16751 || .48270 | 2.07167 || .50441 | 1.98253 || .52650 | 1.89935 | 14 
47| .46171 | 2.16585 || .48306 | 2.07014 |} .50477 | 1.98110 || .52687 | 1.86801 |13 
48) .46206 | 2.16420 | .48342 | 2.06860 || .50514 | 1.97966 || .52724 | 1.89667 | 12 
49| .46242 | 2.16255 || .48378 | 2.06706 || .50550 | 1.97823 |} .52761 | 1.89533 |11 [ 
50! .46277 | 2.16090 || .48414 | 2.06553 || .50587 | 1.97681 |} .52798 | 1.89400 | 10 
51! .46312 | 2.15925 || .48450 | 2.06400 || .50623 | 1.97538 || .52836 | 1.89266, | 9 
2\ .46348 | 2.15760 || .48486 | 2.06247 |) .50660 | 1.97395 || .52873 | 1.89133 | 8 
53| .46383 | 2.15596 || .48521 | 2.06094 || .50696 | 1.97253 || .52910 | 1.89000 | 7 
54| .46418 | 2.15432 || .48557 | 2.05942 || .50733 | 1.97111 || .52947 | 1.88867 | 6 
55| .46454 | 2.15268 || .48593 | 2.05790 |} .50769 | 1.96969 || .52985 | 1.88734 | 5 
56) .46489 | 2.15104 || .48629 | 2.05637 .50806 | 1.96827 || .58022 | 1.88602 | 4 
57| .46525 | 2.14940 || .48665 | 2.05485 || .50843 | 1.96685 |) .53059 | 1.88469 | 3 
58| .46560 | 2.14777 || .48701 | 2.05333 || .50879 | 1.96544 || .58096 1.88837 | 2 
59| .46595 | 2.14614 || .48737 | 2.05182 |; .50916 | 1.96402 || .53134 | 1.88205 | 1 
60| 46631 | 2.14451 || .48773 | 2.05030 |; .50953 | 1.96261 || .53171 1.88073 | 0 
Cotang| Tang ||Cotang; Tang |,Cotang| Tang |/Cotang Tang 
; | a 
65° 64° [yw ABP 62° | 


COTANGENTS. 


TABLE XXVIIL—NATURAL TANGENTS AND 
|, | 28° \| 29° 30° 
| Tang | Cotang | Tang | Cotang |! Tang | Cotang || 
0} .58171 | 1.88073 |) .55431 | 1.80405 | 57785 | 1.73205 | 
1} .53208 | 1.87941 || .55469 | 1.80281 || .57774 | 1.73089 | 
2) .53246 | 1.87809 || .55507 | 1.80158 |; .57813 | 1.72973 
3| .53283 | 1.87677 || .55545 | 1.80084 || .57851 | 1.72857 | 
4| .538320 | 1.87546 || .55583 | 1.79911 || .57890 | 1.72741 
5} .58858 | 1.87415 || .55621 | 1.79788 || .57929 | 1.72625 
6| .533895 | 1.87283 || .55659 | 1.79665 |; .57968 | 1.72509 
7| .584382 | 1.87152 |} .55697 | 1.79542 |; .58007 | 1.72893 
8| .53470 | 1.87021 || .557386 | 1.79419 || .58046 | 1.72278 
9} .538507 | 1.86891 || .55774 | 1.79296 || .58085 | 1.72168 
10| .53545 | 1.86760 || .55812 | 1.79174 |, .58124 | 1.72047 
11} .53582 | 1.86630 || .55850 | 1.79051 || .58162 | 1.71932 | 
12} .53620 | 1.86499 || .55888 | 1.78929 || .58201 | 1.71817 
18| .538657 | 1.86869 |; .55926 | 1.78807 |! .58240 | 1.71702 
14} .53694 | 1.86289 || .55964 | 1.78685 || .58279 | 1.71588 
15| .53732 | 1.86109 || .56003 | 1.78563 || .583818 | 1.71473 
16} .538769 | 1.85979 |; .56041 | 1.78441 || .58857 | 1.71358 | 
17| .53807 | 1.85850 || .56079 | 1.78319 || .583896 | 1.71244 
18| .53844 | 1.85720 || .56117 | 1.78198 |! .58435 | 1.71129 
19} .58882 | 1.85591 || .56156 | 1.78077 || .58474 | 1.71015 
20| .53920 | 1.85462 |} .56194 | 1.77955 |; .58513 | 1.70901 
21| .53957 | 1.85333 || .56282 | 1.77834 || .58552 | 1.70787 
22| .538995 | 1.85204 || .56270 | 1.77713 || .58591 | 1.70673 
23| .54032 | 1.85075 || .56809 | 1.77592 || .58631 | 1.70560 
24| .54070 | 1.84946 || .56347 | 1.77471 || .58670 | 1.70446 
25; .54107 | 1.84818 || .563885 | 1.77351 || .58709 | 1.70382 
25} 54145 | 1.84689 || .56424 | 1.77280 || .58748 | 1.70219 
27| .54183 | 1.84561 || .56462 | 1.77110 |) .58787 | 1.70106 
28} .54220 | 1.84433 || .56501 | 1.76990 |; .58826 | 1.69992 
29! .54258 | 1.84305 || .56639 | 1.76869 || .58865 | 1.69879 
80} .54296 | 1.84177 |} .56577 | 1.76749 |; .58905 | 1.69766 
31| .54333 | 1.84049 || .56616 | 1.76629 || .58944 | 1.69653 
82| .54371 | 1.838922 || .56654 | 1.76510 || .58983 | 1.69541 
33| .54409 | 1.88794 || .56693 | 1.76390 |! .59022 | 1.69428 
84| .54446 | 1.83667 || .56731 | 1.76271 |: .59061 | 1.69316 
85| .54484 | 1.83540 || .56769 | 1.76151 |; .59101 | 1.69203 
36| .54522 | 1.838413 |; .56808 | 1.76032 |; .59140 | 1.69091 
87| .54560 | 1.83286 || .56846 | 1.75913 || .59179 | 1.68979 
38|' 54597 | 1.83159 || .56885 | 1.75794 || .59218 | 1.68866 
39| .54635 | 1.83033 || .56923 | 1.75675 |’ .59258 | 1.68754 
40} .54673 | 1.82906 || .56962 | 1.75556 |: .59297 | 1.68643 
41; .54711 | 1.82780 || .57000 | 1.75487 |' .59336 | 1.68531 
42| .54748 | 1.82654 || .57039 | 1.75319 || .59376 | 1.68419 
43| .54786 | 1.82528 || .57078 | 1.75200 || .59415 | 1.68308 
44} .54824 | 1.82402 |; .57116 | 1.75082 |; .59454 | 1.68196 
45| .54862 | 1.82276 || .57155 | 1.74964 || .59494 | 1.68085 
46| .54900 | 1.82150 || .57193 | 1.74846 || .59533 | 1.67974 
47| .54938 | 1.82025 || .57232 | 1.74728 | .59573 | 1.67863 
48; .54975 | 1.81899 || .57271 | 1.74610 |! .59612 | 1.67°752 
49| .55013 | 1.81774 || .57309 | 1.74492 | .59651 | 1.67641 
50) 55051 | 1.81649 || .57348 | 1.74375 |; .59691 | 1.67530 
51! .55089 | 1.81524 || .57886 | 1.74257 || .59730 | 1.67419 
521 .55127 | 1.81399 || .57425 | 1.74140 || .59770 | 1.67309 
53: 255165 | 1.8127 57464 | 1.74022 || .59809 | 1.67198 
| 54| .55203 | 1.81150 |! .57503 | 1.73905 || .59849 | 1.67088 
55| .55241 | 1.81025 || .57541 | 1.73788 || .59888 | 1.66978 
56| .55279 | 1.80901 || .57580 ; 1.73671 |! 159928 | 1.66867 
57| .55317 | 1.80777 || .57619 | 1.78555 |! .59967 | 1.66757 
58) .55855 | 1.80658 | .57657 | 1.73438 .60007 | 1.66647 
59| 55393 | 1.80529 || .57696 | 1.733821 60046 | 1.66538 
60) .55431 | 1.80405 || .57785 | 1.78205 | .60086 | 1.66428 
\Cotang| Tang |,Cotang: Tang | Cotang| Tang 
, | | 


61° 


EEE Ee eRe cee OR ee RR ee ss 
465 


60° 


59° 


58° 


31° ; 
Tang | Cotang 
60086 | 1.66428 | 60 
.60126 | 1.66318 | 59 
60165 | 1.66209 |58 
.60205 | 1.66099 | 57 
60245 | 1.65990 | 56 
60284 | 1.65881 |55 
60824 | 1.65772 | 54 
.60364 | 1.65663 |53 
60403 | 1.65554 | 52 
60448 | 1.65445 |51 
.60483 | 1.65337 | 50 
60522 | 1.65228 | 49 
60562 | 1.65120 | 48 
60602 | 1.65011 | 47 
.60642 | 1.64903 | 46 
.60681 | 1.64795 | 45 
60721 | 1.64687 | 44 
60761 | 1.64579 | 43 
60801 | 1.64471 | 42 
60841 | 1.64363 | 41 
60881 | 1.64256 | 40 
60921 | 1.64148 |39 
.60960 | 1.64041 |38 
.61000 | 1.63934 | 37 
.61040 | 1.63826 |36 
61080 | 1.63719 | 35 
.61120 | 1.63612 | 34 
.61160 | 1.63505 | 33 
.61200 | 1.63898 | 32 
61240 | 1.63292 | 31 
61280 | 1.63185 | 30 
.61820 | 1.63079 | 29 
.61860 | 1.62972 | 28 
.61400 | 1.62866 | 27 
.61440 | 1.62760 | 26 
61480 | 1.62654 |25 
61520 | 1.62548 |24 
.61561 | 1.62442 |23 
61601 | 1.62336 | 22 
.61641 | 1.62230 | 21 
.61681 | 1.62125 | 20 
61721 | 1.62019 |19 
61761 | 1.61914 |18 
61801 | 1.61808 |17 
61842 | 1.61703 |16 
61882 | 1.61598 | 15 
61922 | 1.61493 |14 
.61962 | 1.61888 |13 
62003 | 1.61283 112 
62043 | 1.61179 |11 
62083 | 1.61074 | 10 
.62124 | 1.60970 | 9 
62164 | 1.60865 | 8 
62204 | 1.60761 | 7 
.62245 | 1.60657 | 6 
62285 | 1.60553 | 5 
62325 | 1.60449 | 4 
62366 | 1.60845 | 3 
62406 | 1.60241 | 2 
62446 | 1.60137 | 1 
.62487 | 1.60033 | 0 

Cotang; Tang 


1 
| 
| 


TABLE XXVIIIL_—NATURAL TANGENTS AND COTANGENTS., 


39° 


li 54° 


“4 | $36 84° T 35° 
Tang | Cotang || Tang | Cotang || Tang | Cotang : Tang | Cotang 
0| .62487 | 1.60033 || .64941 | 1.53986 .67451 | 1.48256 -(0021 | 1.42815 | 60 
1) .62527 | 1.59930 ||} .64982 | 1.53888 67493 | 1.48163 || .70064 | 1.42796 | 59 
2| .62568 | 1.59826 || .65024 | 1.53791 .67536 | 1.48070 -(0107 | 1.42638 |58 
3} .62608 | 1.59723 || .65065 | 1.53693 67578 | 1.47977 || 270151 | 1.42550 |57 
4; .62649 | 1.59620 .65106 | 1.53595 -67620 | 1.47885 || .70194 | 1.42462 156 
5] .62689 | 1.59517 || .65148 | 1.53497 || “67663 1.47792 || .70288 | 1.49874 |55 
6} .62730 | 1.59414 || .65189 | 1.53400 || .67705 | 1.47699 || “70281 1.42286 | 54 
“| .62770 | 1.59311 65231 | 1.53802 || .67748 | 1.47607 || .70325 | 1.42198 | 53 
8} .62811 | 1.59208 65272 | 1.58205 67790 | 1.47514 || .70868 | 1.42110 152 
9} .62852 | 1.59105 65814 | 1.58107 67832 | 1.47422 || .70412 | 1.42022 | 51 
10} .62892 | 1.59002 65355 | 1.58010 || .67875 | 1.47330 || 70455 | 1.41934 |50 
11} .62933 | 1.58900 65397 | 1.52913 | .67917 | 1.47238 || .70499 | 1.41847 | 49 
12} .62973 | 1.58797 65438 | 1.52816 | .67960 | 1.47146 || .70542 | 1.41759 |48 
13} .63014 | 1.58695 65480 | 1.52719 | .68002 | 1.47053 || .70586 | 1.41672 | 47 
14| .68055 | 1.58593 65521 | 1.52622 68045 | 1.46962 || .70629 | 1.41584 | 46 
15| .63095 | 1.58490 || .65563 | 1.52525 .68088 | 1.46870 (0673 | 1.41497 | 45 
16| .63136 | 1.58388 | .65604 | 1.52429 .68130 | 1.46778 .VOT17 | 1.41409 | 44 
17; .63177 | 1.58286 65646 | 1.52332 68173 | 1.46686 || .70760 | 1.41322 | 43 
18} .68217 | 1.58184 65688 | 1.52235 | .68215 | 1.46595 || 70804 | 1.41935 | 49 
19; 63258 | 1.58083 || .65729 | 1.52139 || .68958 | 1.46503 70848 | 1.41148 | 41 
20; .638299 | 1.57981 65771 | 1.52043 | .68301 | 1.46411 |) .70891 | 1.41061 | 40 
21] .63340-| 1.57879 |) .65813 | 1.51946 ‘| 68343 | 1.46320 70935 | 1.40974 |39 
22) .63330 | 1.57778 || .65854 | 1.51850 || |68886 | 1.46299 70979 | 1.40887 | 38 
23| .63421 | 1.57676 |) 165896 | 1.51754 || -68429 | 1.46137 -71023 | 1.40800 | 37 
24) .63462 | 1.57575 || .65938 | 1.51658 || .68471 | 1.46046 -71066 | 1.40714 | 36 
25| .63503 | 1.57474 || .65980 | 1.51562 || .68514 | 1.45955 || .71110 | 1.40697 135 
26} .68544 | 1.57372 .66021 | 1.51466 || .68557 | 1.45864 71154 | 1.40540 | 34 
27} .63584 | 1.57271 |) .66063 | 1.51370 |} -68600 | 1.45773 |! .71198 | 1.40454 |33 
28) .63625 | 1.57170 || .66105 | 1.51275 || .68642 | 1.45689 71242 | 1.40367 | 32 
29] .63666 | 1.57069 || .66147 | 1.51179 || .68685 | 1.45599 71285 | 1.40281 | 31 
30} .63707 | 1.56969 || .66189 | 1.51084 || /68728 | 1.45501 71829 | 1.40195 | 80 
31} .63748 | 1.56868 || .66230 | 1.50988 || .68771 | 1.45410 || .71873 | 1.40109 |29 
32} .63789 | 1.56767 || .66272 | 1.50893 .68814 | 1.45320 71417 | 1.40022 | 28 
33} .63830 | 1.56667 || .66314 | 1.50797 || |68857 | 1.45999 71461 | 1.389936 | 27 
34| .63871 | 1.56566 |} .66356 | 1.50702 || .68900 | 1.45189 || .71505 | 1.39850 |26 
35} .63912 | 1.56466 || .66398 | 1.50607 || 68942 | 1.45049 71549 | 1.39764 | 25 
36} .63953 | 1.56366 | .66440 | 1.50512 || .68985 | 1.44958 71593 | 1.39679 | 24 
37| .68994 | 1.56265 || .66482 | 1.50417 || /¢9028 | 1.44868 71637 | 1.389593 | 23 
38| .64035 | 1.56165 || .66524 | 1.50322 || 69071 | 1 44778 71681 | 1.39507 | 22 
39} .64076 | 1.56065 || .66566 | 1.50228 |} 69114 | 1.44688 11725 | 1.39421 | 21 
40) .64117 | 1.55966 || .66608 | 1.50138 || 69157 | 4 44598 || .71769 | 1.39336 | 20 
41} .64158 | 1.55866 |} .66650 | 1.50038 || .69200 | 1.44508 71813 | 1.89250 | 19 
42) .64199 | 1.55766 || .66692 | 1.49944 .69243 | 1.44418 71857 | 1.39165 |18 
43} .64240 | 1.55666 || .66734 | 1.49849 || |69286 | 1144329 71901 | 1.39079 |17 
44) .64281 | 1.55567 || .66776 | 1.49755 || 69399 | 1.44939 71946 | 1.38994 |16 
45} .64322 | 1.55467 || .66818 | 1.49661 || 69372 | 1144149 71990 | 1.38909 | 15 
46| .64363 | 1.55368 || .66860 | 1.49566 || |69416 | 1.44060 (2084 | 1.38824 | 14 
47| 64404 | 1.55269 |! .66902 | 1.49472 || |69459 | 1.43970 | .@2078 | 1.387388 |13 
48| .64446 | 1.55170 || .66944 | 1.49378 || |69502 | 1.43881 || 12122 | 1.38653 | 12 
49) .64487 | 1.55071 |) .66986 | 1.49284-|| 169545 | 1.43792 72167 | 1.38568 | 11 
50| .64528 | 1.54972 || .67028 | 1.49190 i} .69588 | 1.437038 72211 | 1.388484 | 10 
51} .64569 | 1.54873 | .67071 | 1.49097 | .69631 | 1.43614 || .7@2255 | 1.38399 | 9 
o2| 64610 | 1.54774 |, .67113 | 1.49003 || /69G75 | 1.43595 || .72299 | 1.88314 | 8 
53| .64652 | 1.54675 .67155 | 1.48909 69718 | 1.43486 || .72844 | 1.38229 | 7 
d4| .64693 | 1.54576 .67197 | 1.48816 || .69761 | 1.43347 | .72888 | 1.38145 | 6 
50| .64734 | 1.54478 .67239 | 1.48722 69804 | 1.43258 || 72432 | 1.38060 | 5 
56} .64775 | 1.54879 67282 | 1.48629 69847 | 1.43169 || .72477 | 1.87976 | 4 
O7) .G4817 | 1.54281 |) .67824 | 1.48536 || |c9s91 | 1.43080 || .@2521 | 1.37891 | 8 
103! .64858 | 1.54183 .67366 | 1.48442 69934 | 1.42992 || .72565 | 1.37807 | 2 
59} .64899 | 1.54085 67409 | 1.48349 || .69977 | 1.42903 || .72610 1377221 
| 60| .64941 | 1.53986 67451 | 1.48256 || .70021 | 1.42815 72654 | 1.387688 | 0 
: Cotang; Tang | Cotang| Tang || Cotang | Tang ||Cotang| Tang | 
57° 56° 55° 


TABLE XXVIII.—_NATURAL TANGENTS AND COTANGENTS. 


~ 


SOA mmwwHol 


= 
cose) 


alll aarti amet ceed coeealll come 
IRON WCOre 


| Tang 


36° 
72654 | 1 
72699 | 1 


Cotang 


"1.37638 | 
87554 


37° 


38° 


3 


ge 


Tang 


5 | 1.32704 


53° 


| 


| 
{ 


5 1 
.72743 | 1.37470 | 5447 | 1.382514 | 
(2788 | 1.37386 | 5492 | 1.32464 | 
72832 | 1.37302 || .75538 | 1.82384 | 
£72877 | 1.87218 || .75584 | 1.382804 
72921 | 1.37184 || .75629 | 1.32224 
.72966 | 1.37050 || .75675 | 1.32144 
.738010 | 1.86967 || .7572 1.32064 | 
.73055 | 1.36883 || .75767 | 1.31984 
.73100: | 1.36800 || .75812 | 1.31904 
78144 | 1.36716 || .75858 | 1.31825 | 
.73189 | 1.86633 || .75904 | 1.81745 
78234 | 1.36549 || .75950 | 1.31666 
73278 | 1.36466 || .75996 | 1.31586 
.738323 | 1.36383 || .76042 | 1.31507 
43368 | 1.36300 || .76088 | 1.31427 
.78413 | 1.86217 || .761384 | 1.381848 
T3457 | 1.36134 |; .76180 | 1 31269 
.73502 | 1.36051 || .76226 | 1.31190 
73547 | 1.35968 || .16272 | 1.31110 
73592 | 1.385885 || .763818 | 1.31031 
.73637 | 1.35802 || .76364 | 1.80952 
. 13681 | 1 || .76410 | 1.30873 
13726 | 1 ‘| .76456 | 1.30795 
W377 1.355 . 76502 | 1.30716 
T3816 | 1 | .76548 | 1.30637 
73861 | 1 76594 | 1.380558 
| .73905 | 1 .76640 | 1.380480 
| .%5951 | 1 . 76686 | 1.80401 
73996 | 1 76733 | 1.80323 
- 74041 | 1 76779 | 1.80244 
.74086 | 1 .76825 | 1.30166 
.74131 | 1 (6871 | 1.380087 
.T4176 | 1 .76918 | 1.380009 
(4221 | 1 76964 | 1.29931 
14267 | 1 .77010 | 1.29853 
.14312 | 1 T1057 | 1.29775 
|) .74357 | 1 .77103 | 1.29696 
.74402 | 1 77149 | 1.29618 
74447 | 1 || 77196 | 1.29541 
74492 | 1 77242 | 1.29463 
-74588 | 1 77289 | 1.29885 
74583 | 1. | 77335 | 1.29807 
74628 | 1.33998 || 77382 | 1.29229 
| .74674 | 1.83916 || .77428 | 1.29152 
74719 | 1.388835 || .77475 | 1.29074 
74764 | 1.338754 || .77521 | 1.28997 
.74810 | 1.33673 || .77568 | 1.28919 
| .74855 | 1.388592 || . 77615 1.28842 
| .74900 | 1.38511 || .77661 | 1.28764 
74946 | 1.33430 || .77708 | 1.28687 
“74991 | 1.383849 || . 77754 | 1.28610 
75037 | 1.38268 .77801 | 1.28583 
75082 | 1.388187 || .77848 | 1.28456 
75128 | 1.383107 || .77895 | 1.28379 
75173 | 1.33026 || .77941 | 1.28302 
75219 | 1.382946 || .77988 | 1.28225 
75264 | 1.32865 | 78035 | 1.28148 
75310 | 1.382785 . 78082 | 1.28071 
75355 | 1.32704 || .78129 | 1.27994 
\Cotang| Tang |\Cotang| Tang 


52° 


| Cotang 


82624 | 


|| .78129 
8175 


78269 
78316 
78363 
.78410 
18457 
. 78504 
|| .78551 
78598 
78645 
(8692 
(8039 
78786 
. 78834 
18881 
78928 
68975 
T9022 
79070 | 
TOIL? 
79164 
(9212 
79259 
.79306 
19354 
79401 
T9449 
79496 


79591 
79639 
79686 
T9734 
9781 
79829 
1987 
79924 
79972 
.80020 


80067 
80115 
.80163 
| .80211 

80258 
.80306 


| Cotang 


1 
1 
(8222 | 1.8 
i 
1 
1 


| Tang | Cotang 
~ 78129 | 1.27994 


20917 


27841 


27764 
.27688 
27611 
27589 
27458 | 
27382 
27806 
27230 


271538 
24077 
27001 
26925 
26849 
2677 

26698 
26622 
26546 

26471 
26395 
26319 
26244 
26169 
26093 | 
26018 | 
25943 
25867 


95792 


we 


1 
1 
1 
1 
1 
1 
1 
1 
i 
1 
1 
if 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
79544 | 1, 
a 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
i 
1 
1 


25717 


25642 
25567 
25492 | 
25417 
25343 

25268 
25193 

25118 
25044 | 
24969 


24895 
24820 
24746 | 
24672 
(24597 
24593 


80854 24449 
80402 24375 
80450 24301 
80498 -2ARR7 
80546 .24153 
/80594 24079 

* 80642 .24005 
.80690 | 1.28931 
.80738 | 1.23858 

|] .80786 | 1.23784 
|| .80884 | 1.28710 
.80882 | 1.238637 

| ,80930 | 1.23563 
|| .80978 | 1.23490 
Tang 


| 


wha 


8277 


82825 
82874 
82923 
.82972 
83022 
.83071 
.83120 
.83169 
83218 
.83268 
83317 
83366 
83415 
83465 
83514 
83564 
.83613 
88662 
88712 
83761 
83811 

-83860 

83910 


iCotang | Tang 


50° 


1 


1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
al 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 


1 
18, 
1 


| Tang | Cotang | 
80978 | 
.81027 | 
.81075 
.81123 
.81171 

.81220 

.81268 
.81316 

.81364 
.814138 
.81461 


.81510 
81558 
81606 
81655 
81708 
81752 
| .81800 

| .81849 

81898 
81946 

81995 
“82044 | 
82092 
2141 
82190 
82228 
82287 
82336 
82385 
82434 
82483 
82531 
82580 
82629 
82678 
82727 


/ 

.238490 | 60 
23416 | 59 
23343 | 58 
23270 | 57 
23196 | 56 
23123 | 5D 
.23050 | 54 
22977 [53 
23904 | 52 
22831 | 51 
22758 | 50 
22685 | 49 
22612 | 48 
22589 | 47 
22467 | 46 
22394 | 45 
22321 | 44 
22249 | 43 
22176 | 42 
22104 | 41 
.23031 | 40 
21959. |39 
21886 | 38 
21814 |37 
21742 | 36 
21670 [35 
21598 | 34 
.21526 | 33 
.21454 |32 
.21382 | 31 
.21810 | 30 
21238 | 29 
21166 | 28 
21094 | 27 
21023 | 26 
20951 | 25 
20879 |2 

20808 | 2 

20736 | 22 
20665 | 21 
.20593 | 20 
20522 | 19 
20451 |18 
20379 |17 
20308 | 16 
20237 |15 
20166 |14 
20095 | 13 
20024 | 12 
19958 | 11 
19882 | 10 
.19811 | 9 
19740 | 8 
19669 | 7 
-19599 | 6 
19528 | 5 
19457 | 4 
19387 | 38 
19316 | 2 
19246 | 1 
‘19175 | 0 


= 


DHNIOMBwWoHO! 


TABLE XXVITI.—NATURAL TANGENTS AND COTANGENTS. 

40° 41° | 42° | 43° 
Tang | Cotang || Tang | Cotang |} Tang | Cotang | Tang | Cotang H 
-83910 | 1.19175 || .86929 | 1.15037 || .90040°| 1.11061 || .938252 | 1.07237 | 60 | 
-83960 | 1.19105 |} .86980 | 1.14969 || .90093 | 1.10996 || .93806 | 1.07174 |59 
.84009 | 1.19035 ||} .87031 | 1.14902 |} .90146 | 1.109381 | ,933860 | 1.07112 |58 
.84059 | 1.18964 .87082 | 1.14834 .90199 | 1.10867 || .93415 | 1.07049 | 57 
.84108 | 1.18894 .87133 | 1.14767 .90251 | 1.10802 || .93469 | 1.06987 156 
.84158 | 1.18824 .87184 | 1.14699 .90304 | 1.10737 || .93524 | 1.06925 155 
.84208 | 1.18754 87236 | 1.146382 .90357 | 1.10672 .93578 | 1.06862 | 54 
-84258 | 1.18684 || .87287 | 1.14565 || .90410 | 1.10607 || .93633 | 1.06800 |53 
84307 | 1.18614 || .87338 | 1.14498 || .90463 | 1.10543 || .93688 | 1.06738 |52 
.84357 | 1.18544 .87389 |; 1.14430 .90516 | 1.10478 .93742 | 1.06676 | 51 
.84407 | 1.18474 || .87441 | 1.14363 || .90569 | 1.10414 .93797 | 1.06613 |50 
84457 | 1.18404 || .87492 | 1.14296 || .90621 | 1.10349 | .93852 | 1.06551 | 49 
84507 | 1.18334 || .87543 | 1.14229 || .90674 | 1.10285 |; .93906 | 1.06489 | 48 
84556 | 1.18264 || .87595 | 1.14162 || .90727 | 1.10220 || .93961 | 1.06427-| 47 
-84606 | 1.18194 |) .87646 | 1.14095 || .90781 | 1.10156 || .94016 | 1.06365 | 46 
-84656 | 1.18125 |; .87698 | 1.14028 || .90834 | 1.10091 |! .94071 | 1.06303 | 45 
-84706 | 1.18055 || .87749 | 1.13961 || .90887 | 1.10027 || .94125 | 1.06241 |44 
.84756 | 1.17986 || .87801 | 1.18894 || .90940 | 1.09963 |! .94180 | 1.06179 | 43 
.84306 | 1.17916 || .87852 | 1.13828 || .90993 | 1.09899 || .94235 | 1.06117 | 42 
84856 | 1.17846 || .87904 | 1.13761 || .91046 | 1.09834 || .94290 | 1.06056 | 41 
.84906 + 1.1777’ 87955 | 1.18694 || .91099 | 1.09770 || .94345 | 1.05994 | 40 
84956 | 1.17708 || .88007 | 1.13627 || .91153 | 1.09706 || .94400 | 1.05932 | 39 
-85006 | 1.17638 || .88059 | 1.13561 || .91206 | 1.09642 || .94455 | 1.05870 |38 
-85057 | 1.17569 || .88110 | 1.18494 |} .91259 | 1.09578 || .94510 | 1.05809 | 37 
85107 | 1.17500 || .88162 | 1.13428 || .91313 | 1.09514 || .94565 | 1.05747 136 
85157 | 1.17430 || .88214 | 1.18361 || .91366 | 1.09450 || .94620 | 1.05685 135 
85207 | 1.17361 || .88265 | 1.18295 || .91419 | 1.09386 || .94676 | 1.05624 | 34 
85257 | 1.17292 || .88317 | 1.13228 || .91473 | 1.09322 || .94731 | 1.05562 | 33 
-85308 | 1.17223 || .88369 | 1.18162 |} .91526 | 1.09258 || .94786 | 1.05501 | 32 
.85358 | 1.17154 || .88421 | 1.13096 || .91580 | 1.09195 || .94841 | 1.05439 | 31 
85408 | 1.17085 || .88473 | 1.13029 |} .91633 | 1.09131 || .94896 | 1.05378 |30 
-85458 | 1.17016 || .88524 | 1.12963 || .91687 | 1.09067 || .94952 | 1.65317 |29 
85509 | 1.16947 |) .88576 | 1.12897 || .91740 | 1.09003 |! -.95007 | 1.05255 |28 
.85559 | 1.16878 || .88628 | 1.12831 || .91794 | 1.08940 || .95062 | 1.05194 | 27 
-85609 | 1.16809 || .88680 | 1.12765 || .91847 | 1.08876 || .95118 | 1.05133 | 26 
85660 | 1.16741 || .88782 | 1.12699 || .91901 | 1.08813 || .95173 | 1.05072 | 25 
85710 | 1.16672 || .88784 | 1.12633 || .91955 | 1.08749 |) .95229 | 1.05010 |24 
.85761 | 1.16603 || .88836 | 1.12567 || .92008 | 1.08686 || .95284 | 1.04949 |93 
-85811 | 1.16535 || .88888 | 1.12501 |/ .92062 | 1.08622 || .95340 | 1.04888 | 22 
.85862 | 1.16466 || .88940 | 1.12435 || .92116 | 1.08559 || .95395 | 1.04897 |91 
85912 | 1.16398 |; .88992 | 1.12369 || .92170 | 1.08496 || .95451 | 1.04766 |20 
-85963 | 1.16329 || .89045 | 1.12303 || .92224 | 1.08432 || .95506 | 1.04705 |19 
86014 | 1.16261 |} .89097 | 1.12238 || .92277 | 1.08369 || .95562 | 1.04644 |18 
86064 | 1.16192 || .89149 | 1.12172 |) .92331 | 1.08306 || .95618 | 1.04583 |17 
86115 | 1.16124 || .89201 | 1.12106 || .92385 | 1.08243 |) .95673 | 1.04522 116 
86166 | 1.16056 || .89253 } 1.12041 || .92439 | 1.08179 || .95729 | 1.04461 |15 
86216 | 1.15987 || .89306 | 1.11975 || .92493 | 1.08116 || .95785 | 1.04401 |14 
86267 | 1.15919 || .89358 | 1.11909 || .92547 | 1.08053 || .95841 | 1.04340 113 
.86318 | 1.15851 || .89410 | 1.11844 || .92601 | 1.07990 || .95897 | 1.04279 | 12 
-86368 | 1.15783 || .89463 | 1 11778 || .92655 | 1.07927 | .95952 | 1.04918 111 
86419 | 1.15715 || .89515 | 1.11718 || .92709 | 1.07864 || .96008 | 1.04158 |10 
86470 | 1.15647 || .89567 | 1.11648 || .92763 | 1.07801 ‘| .96064 | 4.04097 | 9 
86521 | 1.15579 |] .89620 | 1.11582.|| .92817 | 1.07738 || .96120 | 1.04036 | 8 
86572 | 1.15511 || .89672 | 1.11517 |) .92°72 | 1.07676 || .96176 | 1.03976 | 7 
.86623 | 1.15443 -89725 | 1.11452 || .92926 | 1.07613 .96232 | 1.08915 | 6 
86674 | 1.153875 || .89777 | 1.11887 || .92980 | 1.07550 || .96288 | 1.08855 | 5 
.86725 | 1.15308 .89830 | 1.11321 .93034 | 1.07487 || .96344 | 1.03794 | 4 
.86776 | 1.15240 89883 | 1.11256 |} .93088 | 1.07425 .96400 | 1.03734 | 3 
.86827 | 1.15172 || .89935 | 1.11191 .93143 | 1.07362 || .96457 | 1.03674 | 2 
.86878 | 1.15104 .89988 | 1.11126 || .93197 | 1.07299 | .96513 | 1.03613 | 1 
.86929 | 1.15037 || 90040 | 1.11061 || .93252 | 1.07237 .96569 | 1.03553 | 0 
Cotang| Tang ||Cotang Tang | Cotang| Tang '|Cotang | Tang 

49° 48° 46° 


TABLE XXVIIL—NATURAL TANGENTS AND COTANGENTS, 


—— 
44° 44° 44° 

, / , ————————___—_—_—__—_—_| / / —————ES 
Tang | Cotang | Tang | Cotang Tang | Cotang 

0 | .96569 | 1.03553 | 60 || 20} .977 1.02355 | 40|} 40) .98843 | 1.01170 
1 | 96625 | 1.03493 | 59 || 21} .97756 | 1.02295 | 39}; 41 .98901 | 1.01112 
29 | 96681 | 1.03433 | 58 || 22| .97813 | 1.02236 | 38/|42| .98958 1.01053 
3 | 96738 | 1.03372 | 57 || 23) .97870 | 1.02176 | 37|| 43} .99016 | 1.00994 
4 | .96794 | 1.03312 | 56 || 24| .97927 | 1.02117 | 36|| 44} .99073 | 1.00935 

| 5 | .96850 | 1.03252 | 55 |) 25 97984 | 1.02057 | 85 |} 45} .99181 | 1.00876 
| 6 | .96907 | 1.03192 | 54 || 26 98041 | 1.01998 | 34|| 46] .99189 | 1.00818 
| 7 | .96963 | 1.03182 | 53 || 27| .98098 1.01939 | 33]/47| .99247 | 1.00759 
8 | .97020 | 1.03072 | 52 || 28) .98155 | 1.01879 | 32 || 48 | .99304 1.00701 
9 | .97076 | 1.03012 | 51 || 29} .98213 | 1.01820 | 31 || 49 | .99362 | 1.00642 
10 | .97133 | 1.02952 | 50 || 30| .98270 | 1.01761 | 30); 50] .99420 1.00583 
11 | .97189 | 1.02892 | 49 || 81] .98327 | 1.01702 | 29))51| .99478 | 1.00525 
12 | .97246 | 1.02832 | 48 || 32| .98384 | 1.01642 | 28||52| .99536 | 1.00467 
13 | .97302 | 1.02772 | 47 || 33| .98441 | 1.01583 | 27||53| .99594 | 1.00408 
14 | .97359 | 1.02713 | 46 || 34] .98499 | 1.01524 | 26|| 54) .99652 | 1.00350 
15 | .97416 | 1.02653 | 45 || 35| .98556 | 1.01465 | 25||55) .99710 | 1.00291 
16 | 97472 | 1.02593 | 44 || 36] .98613 | 1.01406 | 24||56| .99768 | 1.00233 
17 | .97529 | 1.02533 | 43 || 37| .98671 | 1.01347 | 23||57 | .99826 | 1.00175 
18 | .97586 | 1.02474 | 42 || 38) .98728 | 1.01288 22/58 | . 99884 1.00116 
19 | 97643 | 1.02414 | 41 || 39| .98786 | 1.01229 | 21)/59| .99942 | 1.00058 
20 | .977 1.02355 | 40 || 40} .98843 | 1.01170 | 20)) 60 | 1.00000 | 1.00000 

\Cotang| Tang ; Cotang| Tang jes Cotang| Tang 
/ mets a SS eee 
45° 45° 


45° 


| Se wmWROIMD-AIDOO 


et OP 


.00061 


ATO 


Vers. |Ex.sec.|' Vers. |Ex.sec.|| Vers, |Ex. sec. 
0 | .00000 | .00000 |} .00015 | .00015 .00061 | .00061 
1 | .00000 | .00000 .00016 | .00016 .00062 | .00062 
2 .00000 | .00000 .00016 | .00016 -Q0063. | .00063 
38 | .00000 | .00000 -00017 | .00017 .00064 4. .00064 
4 | .00000 | .00000 .00017 | .00017 || .00065 | .00065 
5 | .00000 | .00000 .00018 | .00018 .00066 | .00066 
6 | .00000 | .00000 .00018 | .00018 .00067 | .00067 
7 | .00000 | .00000 .00019 | .00019 .00068 | .00068 
8 | .00000 | .00000 |} .00020 | .00020 .00069 | .00069 | 
9 | .00000 | .00000 .00020 | .00020 .00070 | .00070 | 
10 | .00000 | .00000 .00021 | .00021 .00071 | .90072 | 
.00001 | .00001 .00021 | .Q0021 .00073 | .00073 
.00001 | .00001 .00022 | .00022 00074 | .00074 
00001 | .00001 .00023 | .00023 .00075 | .00075 | 
.00001 |. .00001 .00023 | .00023 .00076 | .00076 
.00001 | .00001 -00024 | .00024 .00077 | .00077 
.00001 | .00001 .00024 | .00024 .00078 | .00078 | 
.00001-| .00001 .00025 | .00025 .00079 | .00079 
.00001 | .00001 -00026 | .00026 .00081 | .00081 
.00002 | .00002 .00026 | .00026 .00082 | .00082 
.00002 | .00002 || .00027 | .00027 || .00083 | .00083 | 
.00002 | .00002 .0002E .00028 .00084 | .00084 
.00002 | .00002 .00028 | .00028 .00085 | .00085 
.00002 | .00002 .00029 | .00029 .00087 | .00087 
.00002 | .00002 . 00030 | .00080 .00088 | .00088 | 
.00003 | .00003 00031 | .00031 .00089 | .00089 | 
.00003 | .00003 .00031 | .00031 .00090 | .00090. | 
.00003 | .00003 .00032 | .00032 .00091 | .00091 | 
.00003 | .00003 |} .00033 | .00033 |} .00093 | .00093 
-00004 | .00004 || .00034 | .00034 |) .00094 | .00094 
.00004 | .00004 || .00034 | .00034 |} .00095 | .00095 
.00004 | .00004 || .00035 | .00035 || .00096 | .00097 
.00004 | .00004 .00036 | .000386 .00098 | .00098 
.00005 | .00005 |} .00037 | .00037 .00099 | .00099 
.00005 | .00005 .00037 | .00037 .00100 | .00100 
.00005 | .00005 .00038 | .00038 .00102 | .00102 
.00005 | .00005 .00039 | .00089 .00103 | .00103 
.00006 | .00006 .00040 | .00040 .00104 | .00104 
.00006 | .00006 .00041 | .00041 .00106 | .00106 
.00006 | .00006 00041 | .00041 .00107 | .00107 | 
.00007 | .00007 .00042 | .00042 .00108 | .00108 
.00007 | .00007 || .00043 | .00043 || .00110 | .00110 
.00007 | .00007 .00044 | .00044 00111 | .00111 
.00008 | .00008 .00045 | .00045 .00112 | .00113 | 
.00008 | .00008 .00046 | .00046 .00114 | .00114 | 
.00009 | .00009 .00047 | .00047 .O0115 |. .00115 
.00009 | .00009 .00048 | .00048 .00117 | .00117 
.00069 | .00009 .00048 | .00048 .00118 | .00118 
.00010 | .00010 .00049 | .00049 .00119 | .00120 
.00010 | .00010 .00050 | .00050 .00121 | .00121 
| .00011 | .00011 .00051 | .00051 .00122 | .00122 
.00011 | .00011 .00052 | .00052 .00124 | .00124 
.00011 | .00011 || .00053 | .00053 4) .00125 | .00125 
-00012 | .00012 || .00054 | .00054 |} .00127 | .00127 
| .00012 | .00012 || .00055 | .00055 || .00128 | .00128 | 
.09013 | .00013 || .00056 | .00056 ||} .09130 | .00130 
.00013 | .00013 |; .00057 | .00057 || .00131 | .00131 
.00014 | .00014 || .00058 | .00058 || .00133 | .00133 | 
| .00014 | .0001% |} .00059 | .00059 || .00134 | .00134 
00015 | .00015 -00060 | .00060 .00136 | .00136 
.00015 | .00015 .00061 .00187 | .00137 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 


Be std 


A 


3° 
4 
Vers. |Ex.sec.| | 
.00137 | .00187 | O > 
.00139 00139 1 
.00140 | .00140 2 
.00142 | .00142 3 
.00143 | .00143 4 
-00145 | .00145 5 
.00146 | .00147 | 6 
.00148 | .00148 | .7 
.00150 | .00150 | 8 
.00151 | .00151 9 
.00153 | .00153 | 10 
.00154 | .00155 | 11 
.00156 | .00156 | 12 
.00158 | .00158 | 1: 
.00159 | .00159 | 14 
.00161 | .00161 | 15 
.00162 | .00163 | 16 
.00164 | .00164 | 17 
.00166 » .00166 | 18 
.00168 | .00168 | 19 
.00169 | .00169 | 20 
-00171.| .00171 | 21 
.00173 | .00173 | 22 
0017 .00175 | 23 
.00176 | .00176 | 24 
.00178 | .00178 | 25 
.00179 | .00180 | 26 
.00181 | .00182 | 27 
.00183 | .00183 | 28 
~ 00185 | .00185 | 29 
.00187 | .00187 | 30 
.00188 | .00189 | 31 
-00190 | .00190 | 32 
.00192 | .00192 | 33 
.00194 | .00194 | 34 
.00196 | .00196 | 35 
.00197 | .00198 | 36 
.00199 | .00200 | 87 
.00201 | .00201 | 38 
.00203 | .00203 | 39 
.00205 | .00205 | 40 
.00207 | .00207 | 41 
.00208 | .00209 | 42 
.00210 | .00211 | 43 
.00212 | .00213 | 44 
.00214 | .00215 | 45 
.00216 | .00216 | 46 
.00218 | .00218 | 47 
.00220 ; .00220 | 48 
.00222 | .00222 | 49 
.00224 | .00224 | 50 
.00226 | .00226 | 51 
.00228 | .00228 | 52 
.00230 | .00280 | 53 
.00232 | .00232 | 54 
.00234 | .00234 | 55 
.00236 | .00236 | 56 
.00238 | .00238 | 57 
-00240 | .00240 | 58 
.00242 | .00242 | 59 
.00244 | .00244 | 60 ~| 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS 


| 

4° 5° 6° ve: 

Yi) | 5 ee ee Fa eee aS, 

| | | 1 

Vers. |Ex.sec.!| Vers. 'Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec.' 
0. | .00244 | .00244 .00381 | .00882 || .00548 | .00551 | .00745 | .00751 
1 | .00246 | .00246 .003883 | .00385 00551 00554 00749 | .00755 
2] .00248 | .00248 || .00886 | .00387 .00554 | .00557 00752 | .00758 
3 | .00250 | .00250 .00388 -| .00390 .00557 | .00560 .00756 | .00762 
4} 00252 | .00252 .00891 .00392 .00560 | .00563 || .00760 | .00765 
5 |) .00254 |-.00254 || .00393 | .00895 .00563 | .00566 .00763 | .00769 
6 | .00256 | .00257 .00396 | .00897 || .00566 | .00569 00767 | .0077% 
7) .00258 | .00259 .00898 | .00400 .00569 | .00573 .00770 | .00776 
8 | .00260 | .00261 .00401 | .00403 00572 | .00576 | 00774 | .00780 
9 | .00262 | .00263 .00404 | .00405 || .00576 | .00579 || .00778 | .00784 
10 | .00264 | .00265 .00406 | .00408 || .00579 | .00582 00781 | .00787 
|} 11 | .00266 | .00267 .00409 | .00411 .00582 | .00585 | 00785 | .00791 
12 | .00269 | .00269 .00412 | .00413 .00585 | .00588 || .00789 | .00795 
13 | .00271 | .00271 .00414 | .00416 .00588 | .00592 .00792 | .00799 
14 | .00273 | .00274 .00417 | .00419 00591 .00595 .00796 | .00802 
15 | .00275 | .00276 .00420 | 00421 00594 |..00598 || .00800 | .00806 
16 | .00277 00278 .00422 | .00424 .00598 | .00601 .00803 | .00810 
ote ee .00280 .00425 | .00427 .00601 | .00604 00807 | .00813 
18 |. .00281 | .00282 .00428 | .00429 .00604 | .00608 .00811 | .00817 
19 | .00284 | .00284 .00430 | .00482 .00607 | .00611 || .00814 | .00821 
20 | .00286 | .00287 .00483 | .00485 .00610 | .00614 .00818 | .00825 
21 | .00288 | .00289 .00436 | .00488 .00614 | .00617 .00822 | .00828 
22 | .00290 | .00291 .00438 | .00440 .00617 | .00621 ||} .00825 . 00832 
23 | .00293 | .00293 .00441 | .00443 00620 | .00624 || .00829 | .00836 
24 | .00295 | .00296 00444 | .00446 || .00623 | ..00627 .00883 | .00840 
25 | .00297 | .00298 00447 | .00449 .00626 | .00630 .60837 | .00844 
26 | .00299 | .003800 .00449 | .00451 .00630 | .00634 | .00840 | .C0848 
27 | .00301 | .00302 || .00452 | .00454 .00633 | .00637 | .00844 | .00851 
28 | .003804 | .003805 .00455 | .00457 .00636 | .00640 .00848 | . 00855 
29 | .00306 | .00307 .00458 | .00460 .00640 | .00644 .00852 | .00859 
80 | .09308 | .00309 || .00460 "| .00463 .00643 | .00647 || .00856 | .00863 
31 | .00811 | .00312 .00463 | .00465 00646 | .00650 |} .00859 | .00867 
382 | .00318 | .003814 .00466 | .00468 .00649 | .00654 || .00863 | .00871 
83 | .00315 | .00316 .00469 | .00471 || .00653 | .00657 || .00867 | 00875 
34 | .00317 | .00318 .00472 | .00474 .00656 | .00660 || .00871 .00878 
39 | .00320 | .00321 .00474 | .00477 .00659 | .00664 || .00875_ | 00882 
36 | 00322 | 00323 || .00477 | :00480 || .00663_| .00667 || .00878 | .00886 
37 | .00824 | .00326 .00480 | .60482 .00666 | .00671 .00882 | .00890 
88 | .00327 | .00328 || .00483 | .00485 .00669 | .00674 || .00886 | .00894 
39 | .00329 | .00380 .00486 | .00488 00673 | .00677 || .00890 | .00898 
40 | .00332 } .00333 .00489 | .00491 .00676 | .00681 .00894 | .00902 
41 .003834 | .00335 .00492 | .00494 .00680 | .00684 || .00898 | .00906 
42 | .003836 | .00337 00494 | .00497 00683 | .00688 || .00902 | .00910 
43 | .00339 | .00340 .00497 | .00500 .00686 | .00691 | 00906 | .00914 
4) .00341 | .00342 .00500 | .00503 00690 | .00695 || .00909 | .00918 
45 | .00343 | .00345 || .005038 | .00506 .00693 | .00698 .00913 | .00922 
A6 | .00346 | .00347 .00506 | .00509 00697 | .00701 |} .00917 | .00926 
47 | .00348 | .003850 .00509 | .00512 00700 | .00705 .00921 | .00930 
48 | .00351 .00352 || .00512 | .00515 00703 | .00708 || .00925 , .00934 
49 | .003853 | .00354 00515 | .00518 || .00707 | .00712 |} .00929 | .00938 
50 | .00856 | .00857 .00518 | .00521 00710 | .00715 || .00933 | .00942 
51 | .00358 | .00359 .00521 | .00524 00714 | .00719 || .00937 | .00946 
2 | .00361 00862 || .00524 | .00527 00717 | .00722 .00941 .00950 
53 | .00363 | .00364 | 00527 | .00530 |} .00721 00726 00945 | .00954 
| 54 | 100365 | :00367 || .00530 | 00533 || .60724 | .00730.| .00949 | .00958 
| 5d | .00368 | .00869 .0053¢ .00536 .00728 | .00733 || .009538 | .00962 
| 56 | 00370 | .00372 || -00536 | |00539 || :00731 | .00737 || 00957 | .00966 
57 | .00373 | .00374 || .00539 | .00542 || .00735 | .00740 || .00961 | .00970 
58 | .00375 | .00377 00542) .00545 .00738 | .00744 .00965 | .00975 
59 | .00878 | .00379 .00545 | .00548 00742 | .00747 | .00969 | .00979 
| 60 | .00381 | .00382 .00551 00745 | .00751 |! .00983 


00548 


00978 


Sequnceeus| 


{ 8° 

a | 

Vers. |Ex. sec 
0 | .00973 | .00983 
1 | .00977 | .00987 
2} .00981 | .00991 | 
3 | .00985 | .00995 
4 | .00989 | .00999 | 
5 | .00994 | .01004 
6 | .00998 | .01008 
7 | .01002 | .01012 
8 | .01006 ; .01016 
9 | .01010 | .01020 | 
10 | .01014 | ,01024 
11 | .01018 | .01029 | 
12 | .01022 | .01033 | 
18 | .01027 | .01037 
14 | .01031 | .01041 
15 | .01035 | .01046 | 
16 | .01039 | .01050 
17 | .01043 | .01054 
18 | .01047 | .01059 
19 | .01052 | .01063 | 
20 | .01056 | .01067 
21 | .01060 | .0107 
22 | .01064 | .01076 | 
23 | .01069 | .01080 
24 | .01073 | .01084 | 
25 | .01077 | .01089 
26 | .01081 | .01093 | 
27 | .01086 | .01097 | 
28 | .01090 | :01102 
29 | .01094 | .01106 | 
30 | .01098 | .01111 | 
31 | .01103 | .01115 | 
82 | .01107 | .01119 | 
33 | .01111 | .01124 | 
34 | .01116 | .01128 
35 | .01120 | .01133 
36 | .01124 | .01137 | 
37 | .01129 | .01142 | 
38 | .01133 | .01146 
89 | .01137 | .01151 
40 | .01142 | .01155 
41 | .01146 | .01160 
42 | .01151 | .01164 
43 | .01155. | .01169 
44 | .01159 | .01173 
45 | .01164 | .01178 | 
46 | .01168 | .01182 | 
47 | .01173 | .01187 
48 | .01177 | .01191 
49 | .01182 | .01196 
50 | .01186 | .01200 
51 | .01191 | .01205 | 
52 | .01195 | .01209 | 
53 | .01200 | .01214 
54] .01204 | .01219 | 
55 | .01209 | .01223 
56 | .01213 | .01298 
57 | .01218 | .01233 | 
58 | .01222 | .01237 | 
59 | .01227 | .01242 | 
60 | .01231 | .01247 | 


9° 10° 11° 
Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex. sec.| 
|| .012381 | .01247 |} .01519 | .01543 .018387 | .01872 
.01236 | .01251 .01524 | .01548 .01843 | .01877 
.01240 | .01256 || .01529 | .01553 .01848 | .01883 
.01245 | .01261 .015384 | .01558 .01854 | .01889 
.01249 | .01265 .01540 | .01564 .01860 | .01895 | 
.01254 | .01270 -01545 | 01569 .01865 | .01901 
.01259. | .0127 -01550 | .01574 .01871 | .01906 
.01263 | .01279 .01555 | .01579 .01876 | .01912 
.01268 | .01284 .01560 | .01585 .01882 | .01918 
.01272 | .01289 .01565 | .01590 .01888 | .01924 
01277 | .01294 .01570 | .01595 .01893 | .01980 
-01282 | .01298 || .01575 | .01601 || .01899 | .01936 
.01286 | .01303 .01580 | .01606 .01904 | .01941 
.01291 | .01308 .01586 | .01611 .01910 | .01947 
.01296 | .01313 .01591 | .01616 -01916 | .01953 
-01300 | .01318 || .01596 | .01622 || .01921 | _01959 Ve J 
.01305 | .01322 -01601 | .01627 .01927 | .01965 
.01310 | .01327 -01606 | .01633 .01933 | .01971 
.01314 | .01332 .01612 | .01638 .01939 | .01977 - 
-01319 | .01837 .01617 | .01643 .01944 | .01983 
-O1324 | .01342 || .01622 | .01649 || .01950 | _01989 
-01329 | .01346 |) .01627 | .01654 || .01956 | .01995 
.01833 | .01351 .01632 | .01659 || .01961 | .02001 
.01338 | .01356 .01638 | .01665 .01967 | .02007 
-01343 | .01361 || .01643 | .01670 || .01973 | (02013 
.01348 | .01366 -01648 | .01676 || .01979 | 02019 
.01352 | .01371 .01653 | .01681 .01984 | .02025 
.01357 | 01376 .01659 | .01687 .01990 | .02031 
.01362 | .01381 .01664 | .01692 .01996 | .02037 
.01367 | .01386 .01669 | .01698 .02002 | .02043 
01871 | .01391 || .01675 | .01703 || .02008 | |02049 
-01376 | .01395 || .01680 | .01709 || .02013 | .o2055 
.01381 | .01400 .01685 | .01714 .02019 | .02061 
.01386 | .01405 .01690 | .01720 :02025 | .02067 
.01391 | .01410 -01696 | .01725 || .02031 | .02073 
.01396 | .01415 -O1701 | .01731 || .02037 | .o2079 
.01400 | .01420 .01706 | .01736 .02042 | .02085 
.01405 | .01425 .01712 | .01742 .02048 | .02091 
-01410 | .01430 || .01717 | .01747 || .02054 | |02097 
.01415 | .01435 01723 | .01753 .02060 | .02103 
-01420 | .01440 |) .01728 | .01758 || .02066 | .02110 
-01425 | .01445 || .01733 | .01764 {| .02072 | .02116 
-01430 | .01450 || .01739 | .01'769 || .02078 | .02122 
.01435 | .01455 .01744 | .0177 .02084 | .02128 
.01489 | .01461 .01750 | .01781 -02090 | .02134 
-01444 | .01466 .01755 | .01786 || .02095 | .02140 
.01449 | .01471 -01760 | .01792 || .02101 | .02146 
-01454 | *01476 ||. .01766 | .01798 | .02107 | .02158 
-01459 | .01481 || .0177 .01803 .02113 | .02159 
.01464 | .01486 || .01777 | .01809 || .02119 | .02165 
-01469 | .01491 || .01782 | .01815 .02125 | .02171 
.01474 | .01496 .01788 | .01820 || .02131 | .02178 | 
.01479 | .01501 .01798 | .01826 .02137 | .021484 
.01484 | .01506 .01799 | .01832 || .02143 | .02190 
.01489 | .01512 01804 | .01837 || .02149 | .02196 
-01494 | 01517 .01810 | .01843 || .02155 | .02203 
-01499 | .01522 .01815 | .01849 || .02161 | .02209 | 
-01504 | .01527 .01821 | .01854 .02167 | .02215 
-01509 | .01532 -01826 | .01860 || -02173 | .02221 
-01514 | .01537 .01832 | .01866 | -02179 | .02228 
.01519 | .01543 .01837 | .01872 || .02185 02234 


wcomewHe | 


| 


ee eae 


k 


esl 


THe COD 


e 
Oo 


Mm-Io 


12° 


14° 


15° 


02447 


.02508 


| 02453 | .02515 
02459 | .02521 
02466 | .02528 
02472 | .02535 
02479 | .02542 
102485 | .02548 
(02492 | .02555 
02498 | .02562 
.02504 | .02569 
02511 | .02576 
02517 | .02582 
02524 | .02589 
.02530 | .02596 
.02537 | .02603 | 
02543 | 02610 | 
02550 | .02617 
02556 | .02624 
02563 | .02630 


02845 
02852 
02859 
02866 


| .02873 
| .02880 
| .02887 


02894 
.02900 


.02907 
.02914 
.02921 
.02928 
.02935 
02942 
.02949 
.02956 
.02963 
.02970 


02928 
.02936 
02943 
02950 
02958 
02965 
02972 
.02980 
.02987 
.02994 
03002 
03009 
03017 
03024 
03032 
03039 

03046 
03054 
.03061 


Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. 
.02185 | .02234 || .02563 | .02630 || .02970 
-02191 | .02240 .02570 | .02637 || .02977 
.02197:| .02247 || .02576 | .02644 02985 
| .02203 | .02253 .02583 | .02651 .02992 
.02210 | .02259 .02589 | .02658 .02999 
02216 | .02266 || .02596 | .02665 .08006 
02222 | .02272 || .02602 | .02672 .03013 
02228 | .02279 || .02609 | .02679 . 03020 
.02234 | .02285 || .02616 C2686 .08027 
.02240 | .02291 || .02622 | .02693 .08084 
02246 | .02298 |} .02629 | .02700 .08041 
| .02252 | .023804 .026385 | .02707 .03048 
.02258 | .02311 || .02642 | .02714 || .08055 
.02265 | .023817 || 2649 | .02721 .08063 
02271 | .02323 || .02655 | .02728 || .03070 
.02277 | .02330 || .02662 | .02735 .03077 
.022838 | .02336 .02669 | .02742 || .03084 
.02289 | .02343 .02675 | .02749 .038091 
.02295 | .02349 || .02682 | .02756 .03098 
02302 | .023856 || .02689 | .02763 || .08106 
.02808 | .02862 |} .02696 | .02770 || .03113 
.02814 | .02369 || 02702 | .02777 |) .03120 
0232 .02375 02709 | .0272°4 .03127 
.02327 | .02382 02716 | 02791 .08134 
.02333 | .02388 -02722 | .02799 .08142 
02339 | .02395' || .02729 | .02806 .08149 
02345 | .(2402 || .02736 | .02813 || .08156 
.02352 | .02408 || .02743 | .02820 || .03163 
.02358.| .02415 || .02749 | .02827 038171 
.02364 | .02421 || .02756 | .02834 08178 
.02370 | .02428 .02763 | .02842 |) .03185 
.02377 | .02485 || .02770 | .02849 .03193 
02383 | .02441 02777 .02856 || .03200 
| .02889 | .02448 || .02783 | .02863 || .038207 
02896 | .02454 .02790 | .02870 2038214 
02402 | .02461 || .02797 | .02878 08222 
.02408 | .02468 || .02804 | .02885 .03229 
y | 02415 | .02474 .02811 | .02892 .038236 
02421 | .02481 || .02818 | .02899 || .08244 
02427 | 02488 || .02824 | .02907 .03251 
02434 | .02494 || .028381 | .02914 || .08258 
.02440 | .02501 .028388 | .02921 .03266 


| .032738 
.03281 
03288 
.08295 
03303 
03310 
.03318 
.03325 
03333 
.03340 
03347 
-U8500 
| .0383862 
.08370 
03377 
.03385 


03355 


03392 


03400 


Ex. sec. | Vers. |Hx. sec. 
03061 03407 | .03528 
.03069 03415 | .03536 
.08076 || .03422 | .08544 
.08084-|| .03480 | .08552 
.03091 .03488 | .03560 


.03099 


.03106 


.03114 
.08121 


.03129 


.03137 
.03144 
03152 


.08159 
.03167 
.08175 


03182 


.03190 
03198. | 


.08205 


08213 | 


03221 
.03228 
.032386 
05244 
.08251 
.08259 
08267 
.08275 
08282 


.03290 || 
.03298 


03306 


.033813 


.03821 
.03329 
03337 
.03345 
.03353 
.03360 
03368 
.03376 
.08384 
03392 


.03400 


.03408 
.03416 
.03424 


03432 


03439 
03447 
.03455 
03468 
08471 
03479 
.08487 


.03495 
03503 


03512 


03407 


.03520 
03528 


.03445 | .08568 
.03453 | .08576 


.03460 | .03584 
.03468 | .03592 


03476 
03483 
.03491 
03498 
03506 
.03514 
03521 
.03529 
03537 
03544 
03552 
03560 


03601 
.08609 


.03617 


.03633 
.03642 
03650 
.03658 
.03666 
03674 
03683 
.03691 


.03567 | .03699 
03575 | .03708 


.03583 
03590 
.03598 
.03606 
.03614 
.03621 | .08758 
.03629 | .08766 
.03637 | .G8774 


.03716 
038724 
03732 
03741 
03749 


.038645 | .03783 
.03653 | .08791 
.03660 | .08799 
.03668 | .03808 
.03676 | .03816 
03684 | .03825 
03692 | .03833 
03699 | .03842 
03707 | .03850 
.03715 | .03858 
.03723 | .08867 
087381 | .08875 
.03739 | .08884 
08747 | .038892 


.08754 | .038901 
.03762 | .03909 


0877 .03918 
0377 .08927 
-03786 | .03935 
.03794 | .03944 


.08802 | .03952 


.03810 | .08961 
.03818 | .03969 
03826 | .03978 
.03834 | .08987 
.03842 | .03995 
.03850 | .04004 
.03858 | .04013 
.03866 | .04021 


03874 | .04030 


473 


03625 | 


<XIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 


Serecmowno| 


16° 


Vers. | 
{ 


Ex. sec.'!| Vers. |/Ex.sec. | Vers 
| | 
0 | 04370 | .04569 || .04894 | 05448 
} | .04378 | .04578 || .04903 | 05458 
2 | .04887 | ,04588 || .04912 | 05467 
3 -04395 | .04597 |, ,04921 05477 |. 
4 | .04404 | .04606 || .04930 | | .05486 
5 .04412 | .04616 || .04939 | | .05496 
6 .04421 | .04625 || 04948 | .05505 | 
fi .04429 | .04635 || .04957 | .05515 | 
8 | 04488 | .04644 || .04967 | .05524 1, 
9 .04446 | .04653 || .04976 05584 : . 
| .04455 | .04663 || .04985 05543 
.04464 | .04672 || .04994 .05553 | 
| .04472 | .04682 || .05003 05562 
| .04481 | .04691 || .05012 | .05572 | 
| .04489 | .04700 || .05021 | | 05582 
| .04498 | .04710 || .05030 | | .05591 
| | .04507 | .04719 || .05089 | .05601 | 
04515 | .04729 || .05048 | 05610 | 
.04524 | .04738 || .05057 | ‘| .05620 | 
.04533 | .04748 || 05067 7 || .05680 | 
.04541 | .04757 || .05076 1} .05639 
.04550 | .04767 || .05085 | .05649 | 
'| .04559 | .04776 || .05094 || 05658 | 
.04567 | .C4786 |! 05103 | | .05668 
.04576 | .04795 |) 05112 05678 
.04585 | .04805 || .05122 | 05687 
.04593 | .04815 |} .05131 | 05697 
1} .04602 | .04824 || .05140 | | 05707 | . 
.04611 | .04834 |} .05149 05716 | . 
|| .04620 | .04848 |] .05158 05726 
04628 | .04853 || .05168 05736 
.04637 | .04€63 || .05177 05746 | 
.04646 | .04872 || .05186 05755 
.04655 | .04882 || .05195 05765 
.04663 | .04891 || .05205 SO57%5 |. 
04672 | .04901 || .05214 05785 |. 
.04681 | .04911 || .05293 05794 | 
.04690 | .04920 || .05232 05804 
.04699 | .04980 || ,05242 05814 
.04707 | .04940 || .05251 05824. | 
.04716 | .04950 || .05260 .05833 | 
.04725 | .04959 || .0527¢ 05843 
i} .0473 .04969 || .05279 '} 05853 
i| .04743 | .04979 || .05288 i| .05863. | 
|| .04752 | .04989 || .05298 05873 
|| .04760 | .04998 || .05307 05882 
.04769 | .05008 |} .05316 .05892 | 
| .04778 | .05018 ||. .05326 05902 
| .04787 | .05028 || .05335 .05912 
.04796 | .05038 || .05344 || .05922 
.04805 | .05047 |) .05354 || .05932 
.04814 | .05057 || .05363 .05942 
'| .04823 | .05067 |! .05373 05951 
‘| .04832 | .05077 || .05382 | . 05961 | 
04841 | .05087 || .05391 05971 
.04850 | .05097 || .05401 05981 
|| -04858 | .05107 || .05410 05991 
|| .04867 | .05116 || .05420 06001 
|| .04876 | .05126 || .05429 06011 
.04885 | .05136 || .05439 .06021 
.04894 | .05146 || .05448 .06031 


TABLE XXIX.—NATURAL VERSED SINHS AND EXTERNAL 


an 


10 


) 
SCmrIaumeIne| 


| | 
06031 | .06418 || .06642 
06041 | .06429 || .06652 
06051 | .06440 || .06663 
06061 | .06452 || .06673 
06071 | .06463 || .06684 
.06081 | 706474 || .06694 
.06091 | .06486 || .06705 
.06101 | .06497 || .06715 
.06111 | .06508 06726 
.06121 | .06520 .06736 
.08131 | .06531 || .06747 
.06141 | .06542 .06757 
.06151 | .06554 || .06768 
.06161 | .06565 || .06778 
06171 | .06577 || .06789 
06181 | .06588 || .06799 
.06191 | .05600 || .0G8i0 
| .06201 | .06611 || .06820 
.06211 | .06622 06831 
06221 | .06634 |} .06341 
06231 | .06645 || .06552 
06241 | .06057 || .06°63 
06252 | .00668 || .06873 
06262 | .03580 || .06884 
06272 | .06691 || .05894 
.06282 | .05703 .06905 
06292 | .0S715 || .06916 
.06302 | .05726 || .06926 
.06312 | .06738 || .06937 
.06323 | .06749 || .06948 
.06333 | .05761 ||} :06958 
.06343 | .03773 || .06969 
.06353 | .06784 || .06980 
06363 | .06796 || .06990 
.06374 | .06807 O7F00L 
.06384 | .06819 || .07012 
.06394 | .06831 |} .07022 
.06404 | .06843 || .07033 
.06415 | .06854 || .07044 
| 06425 | .06866 || .07055 
| .06485 | .06878 .07065 
| .06445 | .06889 07076 
06456 | .06901 || .07087 
.06466 | .06913 || .07098 
.06476 | .06925 || .07108 
.06486 | .06936 || .07119 
06497 | .06948 || .07130 | 
| .06507 | .06960 || .07141 
06517 | .06972 || .07151 
| .06528 | .06984 || .07162 
06538 | .06995 || .07173 
06548 ; .07007 || .07184 
06559 | .07019 || .07195 | 
.06589 | .07031 || .07206 
.06580 | .07043 || .07216 | 


20° 


Vers. |Ex.sec.| 


21° 


SECANTS. 


Vers. |Ex. sec. 


06590 | .07055 || 
“06600 | .07067 || 
06611 | .07079 | 


.06632 


| 
06621 | . 
06642 | . 


07091 
7103 
07115 | 


07227 
0723 

07249 
07260 
07271. | 
07282 | 


.07602 
.07615 
07627 
.07640 
.07652 
.07665 
07677 
.07690 
.07 702 
Oia ks) 
(07725 
07740 | 
97752 
07765 
07778 
.07790 
.07803 
07816 
.07828 
07841 
.07853 


| 


O7904 | 
07915 


SORO2T 4 


OTT 24 
.07735 
.07746 
OTTST 
07769 
07780 
07791 | 
07802 
07814 
07829 
07336 
.07818 
07859 
07870 
07881 


07955 


7393 


079338 


08370 
08383 
083897 
08410 
08423 
08436 


08449 || 


. 08463 


.08476 


08489 || 
.08505 || 


3516 
08529 | 


(08542 || 


.08555 
08569 
. 038582 


08596 || 
.0860.) |} 
08623 || 
08636 |; 


.08410 | .09183 


08422 | .09197 
.08434 | .09211 


.08445 | .09224 | 
.08457 | .09238 
.08469 | .09252 | 
.08481 | .09266 | 
.08492 | .09280 | 
.08504 | .09294 | 
.08516 | .09305 
.08528 | .09823 


.08539 | .09837 


.08551 | .09351 
.08563 | .09365 
08575 | .09379 | 
.08586 | .09398 

| 08598 | .09407 
.O8610 | .09421 
.08622 | .09435 
08634 | .09449 
.08645 | .09464 


| 22° 238 
| = ie , 
| Vers, |Ex sec. | Vers. |Ex. sec.| 
07115 || .07282 | .07853 || .07950 | .08636 | 0 
07126 || .07293 | .07866 || .07961 | .08649 | 1 
07138 || .07303 | .07879 || .07972 | .08663 | 2 
- 07150 || .07314 | .07892 || .07984 | .08676 | 3 
07162 || .07325 | .07904 || .07995 | .08690 | 4 
07174 || .07336 | .07917 || .c8006 | .08703 | 5 
07186 || .07347 | .07930 |} .08018 | .08717 | 6G 
07199 || .07358 | .07943 || .08029 | .08730 | 7 
07211 || .07369 | .07955 || .08041 | .08744 | 8 
f223 || .07380 | .07968. |) .08052 | .08757 | 9 
n935 || .07391 | .07981 || .08064 | .08771 | 10 
07247 || .07402 | .07994 || .08075 | .08%84 | 11 
07259 || .07413 | .08006 || .08086 | .08798 | 12 
0727 07424 | .08019 |} .08098 | .O8811 | 13 
07288 || .07435 | .08032 |} .08109 | .08825 | 14 
07295 || .07446 | .08045 || .08121 | .08839 | 15 
07207 || .07457 | .08058 || .08182 | .08852 | 16 
.07320 || .07468 | .08071 || .08144 | .08866 | 17 
"073232 || .07479 | .08084 || .08155 | .08880 | 18 
07344 || .07490 | .08097 |} .08167 | .08893 | 19 
.07356 || .07501 08109 || .08178 | .08907 | 20 
07368 || .07512 | .08122 || .08190 | .08921 | 21 
07380 || 68135 || .08201 | .08934 | 22 
07393 08148. || .08213 | .08948 | 28 
07405 || 08161 || .08225 | .08962 | 24 
"O7417 || .07556 | .08174 || .08236 | .08975 | 25 
07429 || .07568 | .08187 || .08248 | .08989 | 26 
07442 || .07579 | .08200 || .08259 | .09003 | 27 
07454 |) .07590 | .08213 || .08271 | .09017 | 28 
07466 || .07601 | .08226 || .08282 | .09030 | 29 
07479 || .07612 | .08239 || .08294 | .09044 | 30 
07491 || .07628 | .08252 || .08306 | .09058 31 
07503 || .07634 | .08265 || .08817 | .09072 | 32 
07516 || .07645 | .08278 || .08829 | .09086 | 33 
07528 || .07657 | .08291 || .083840 | .09099 | 34 
“07540 || .07668 | .08305 || .08852 | .09113 | 35 
07553 || .07679 | .083818 || .08864 | .09127 | 36 
07565 || .07690 | .08833 08375 | .09141 | 37 
07578 || .07701 | .08844 || .08887 | .09155 | 38 
07590 || .07713. | ..08357 || .08399 | .09169 | 389 


ATS 


24° 


25° 


26° 


27° 


Vers. | Ex. sec. 


| .08645 
08657 | 


.O8669 


| .08681 
| .08693 


08705 
08717 


| .08728 


.08740 
08752 
. 08764 
0877’ 

.08788 
. 08800 


| .08812 


08824 
-08836 
. 08848 
.O8860 
03872 
03884 
.08896 
-08908 
.08920 
.089382 
.08944 
. 08956 
.08968 
.08980 


08992 | 


.09004 


.09016 
09028 
.09040 
.09052 
.09064 
.09076 


| .09089 


.09101 
99118 
.09125 


.09137 
09149 
-O9161 


.09186 
.09198 
.09210 


- 09222 


.09234 


.09247 | 


. 00259 


.09271 | 


09283 


09296 | 
09308 | 


.09320 


| 09174 | 


09832 | 


09345 


09857 } 


.09369 | 


= | bore? 
| 
Vers. |Ex.sec.|| Vers. |Ex. sec.|| Vers. |Ex. sec. | 
| { 
.09464 || .09369 | .10338 |} .10121 | .11260 || 210899 | 1228371 20 
.09478 |) .09382 | .10353 || .10183 | .11276 || .10913 | .12949 1 
. 09492 .09394 | .10368 || .10146 | .11292 |} 10926 | .12266 | 2 
-09506 |; .09406 | .10383 || .10159 | .11308 || .10989 | .12283 | 3 
.09520 .09418 | .10398 10172 | .11823 |} .10952 | .12299 4 
09535 || .09431 | .10413 |} .10184 | .11339 || .10965 | .12316 | 5 
.09549 09442 | .10428 || .10197 | .11355 || .10979 | .12333 6 
-09563 || .09455 | .10443 || .10210 | .11371 |' .10992 | .12349 | 7 
09577 || .09468 | .10458 || .10223 | .11387 || .11005 | .12366 8 
-09592 || .09480 | .10473 || .10236 | .11403 |; .11019 | .12883 | 9 
-09606 |} .09493 | .10488 |} .10248 | .11419 } 11032 | .12400 | 10 
-09620 || .09505 | .10503 || .10261 | .11435 |) .11045 | .12416 | 11 
.09635 || .09517 | .10518 || .10274 | .11451 || .11058 | .12433 | 12 
-09649 || .09530 | .10533 || .10287 | .11467 || .11072 | .12450 | 13 
09663 |} .09542 | .10549 || .10300 | .11483 || .11085 | .12467:| 14 
-09678 || .09554 | .10564 |] .10313 | .11499 |} .11098 | .12484 | 15 
-09692 |; .09567 | .1057 .10826 | .11515 -11112 | .12501 | 16 
09707 || .09579 | . 10594 |} .10388 | .11531 || .11125 | .12518 | 17 
.09721 -09592 | .10609 || .10351 | .11547 || .11138 | .12534 | 18 
.09735 |) .09604 | .10625 -10864 | .11563 |; .11152 | .12551 | 19 
09750 || .09617 | .10640 |} .10377 | .11579 || .11165 | .12568 | 20 
-09764 |} .09629 | .10655 || .10390 | .11595 || .11178 | .12585 | 21 
09779 .09642 | .10670 -10403 | .11611 || .11192'| .12602 | 22 
09793 || .09654 | .10686 || .10416 | .11627 11205 | .12619 | 23 
-09808 || .09666 | .10701 || .10429 | .11643 |} .11218 | .12636 | 24 
-09822 || .09679 | .10716 || .10442 | .11659 |} .11232 | .12653 | 25 
-09887 |} .09691 | .10781 || .10455 | .11675 |} .11245 | .12670 | 26 
09851 || .09704 | .10747 || .10468.} .11691 || .11259 | .12687 | 97 
-09866 || .09716 | .10762 || .10481 | .11708 || .11272 | .19704 | 98 
.09880 || .09729 | .10777 }} .10494 | .11724 | -11285 | .12721 | 29 
.09895 || .09741 | .10793 .10507 «| .11740 | 11299 | .12788 |}, 380 
-09909 |; .09754 | .10808 || .10520 | .11756 || .11812 | .19755 | 31 
09924 || .09767 | .10824 -10533 | .11772 ||- .11826 | .12772 | 82 
-09939 || .09779 | .10839 |} .10546 | .11789 |) .11389 | .12789 | 33 
-09953 || .09792 | <10854 |! .10559 | .11805 || 11358 | .19807 | 34 
-09968 |} .09804 | .10870 |; .10572 | .11821 || .11366 | .19824 | 35 
-09982 || .09817 | .10885 .10585 | .11838 .11380 | .12841 | 86 
.09997 .09829 | .10901 -10598 | .11854 |! .11893 | .12858 | 37 
-10012 |} .09842 | .10916 |} .10611 | .11870 |} .11407 | .19875 | 38 
- 10026 .09854 | .10982 || .10624 | .11886 | .11420 | .12892 | 39 
-10041 |) .09867 | .10947 |; .10637 | .11903 |} .11434 | .19910 | 40 
-10055 || .09880 | .10963 || .10650 | .11919 |) .11447 | .12907 | 44 
-10071 |; .09592 | .10978 || .10663 | .11936 |} .11461 12944 | 42 
- 10085 09905 | .10994 |} .10676 | .11952 |) .11474 | .12961 | 43 
-10100 |} .09918 | .11009 || .10689 | .11968.|| .11488 | .12979 | 44 
-10115 || .09980 | .11025 |; .10702 | .11985 |} .11501 | .12996 | 45 
10139 |] .09943 | .11041 |} .10715 | .12001 .11515 | .13013 | 46 
10144 || .09955 | .11056 |, .10728 |. .12018 ;| 11528 | .138081 ; 47 
-10159 || .09963 | .11072 |] .10741 |. .12034:|] .11542 | .13048 | 48 
10174 || .09981 | .11087 || .10755 | .12051 .11555 | .18065 | 49 
-10189 || .09993 | .11103 || .10768 | .12067 || .11569 | .13083 | 50 
-10204 || .10006 | .11119 || .10781 | .12084 || .11583 | .13100 | 51 
-10218 |} .10019 |. .11134 || .10%794 | .12100 | .11596 | .18117 | 52 
-10233 |) .10082 | .11150 || .10807 | .12117 || .11610 | .13135 | 53 
-10248 || .10044 | .11166 |] .10820 | .12138 |} .11623 | .13152 | 54 
-10263 |} .10057 | .11181 || .10833 | .12150 || .11637 | .13170 | AS 
-10278 || .10070 | .11197 || .10847 | .12166 |} .11651 | .13187 | 56 
-10293 || .10082 | .11213 |} .10860 | .12183 |! .11664 . 18205 | 57 
-10308 || .10095 | .11229 || .10873 | .12199 |! .11678 .18222 | 58 
-10323 |] .10108 | .11244 || .10886 | .12216 || .11692 | 13240 | 59 
-10338 || .10121 | .11260 || .10899 | .12233 |} .11705 | .13257 | 60 | 


TADLE XXIX._NATURAL VERSED SINES AND EXTERNAL SECANTS. 


29° 


31° 


| 


Vers. |Ex. sec Vers. |Ex. sec.|| 
| | 
0 | .11705 | .18257 |] .12588 | .14385 
4} .11719 | .18275 || .12552 | .14354 
2 | 141733 | 113292 || 119566 11372 
8 | .11746 | .13310 || .12580 4391 
4 | .11760 | .13827 || .12595 ar 
5 | 1177 13345 || .12609 | .14428 
6 | .11787 | .13362 2623 | 14446 
* | .11801 | .13380 || .12687 | .14465 
8 | .11815 | .13398 || .12651-| .14488 
9 | .11828 | .13415 || .12665 | .14502 
10 | 11842 | .13433, || .12679 | .14521 
11 | .11856 | .13451 || .12694 | .14539 
12 | .11870 | .13468 || .12708 | .14558 
18 | 11888 | .13486 || .12722 | .14576 
14 | .11897 | .13504 || .12736 | .14595 
15 | .11911 18521 12750 | .14614 
16 | .11925 3539 || .12765 | .14632 
17 | .11938 “5667 1277 14651 
18 | .11952 | .13575 || 12793 | .14670 
19 | .11966 | .18593 || .12807 | .14689 
20 | .11980 | .13610 |} .12822 | .14707 
21 | .11994 | .13628 || .12836 | .14726 
22 | 12007 | .13646 || .12850 | .14745 
23 | .12021 | .13664 || .12864 | .14764 
24 | .12035 | .13682 |} .12879 | .14782 
25 | .12049 | .13700 || .12893 | .14801 
26 | .12063 | .13718 || .12907 | .14820 
27 | .12077 | .18785 || .12921 | .14889 
28 | ,.12091 | .13753 || .12936 | .14858 
29 | .12104 | .1377 12950 | 14877 
30 | .12118 | .13789 || .12964 | .14896 
31 | .12132 | .13807 || .12979 | .14914 
82 | .12146 | .13825 || .12993 | .14933 
33 | .12160 | .13843 || .13007 | .14952 
34 | .12174 | .18861 || .13022 | .14971 
35 | .12188 | .13879 || .13036 | .14990 
36 | .12202 | .13897 || .13051 | .15009 
37 | .12216 | .13916 || .13065 | .15028 
38 | .12230 | .13934 || .13079 | .15047 
39 | .12244 | .13952 || .18094 | .15066 
40 | .12257 | .13970 || .13108 | .15085 
41 | .12271 | .13988 || .13122 | .15105 
42 | .12285 | .14006 |} .13137 | .15124 
43 | .12299 | .14024 || .18151 | .15143 
44 | .12313 | .14042 |] .13166 | .15162 
45 | .12827 | .14061 || .13180 | .15181 
46 | .12341 | .14079 || .13195 | .15200 
47 | .12355 | .14097 |} .18209 | .15219 
48 | .12869 ) .14115 || .13228 | .15939 
49 | .12388 | .14134 |] .13288 | .15258 
50 | .12897 | .14152 || .18252 | 15277 
51 | .12411 | .14170 || .13267 | .15296 
52 | 12425 | .14188 |] .13281 | .15815 
53 | .12489 | .14207 || .13206 | .15335 
54 | 112454 | .14225 || .18810 | .15354 
55 | .12468 | .14243 || .13825 | .15373 
56 | .12482 | .14262 || .13839 | .15393 
57 | .12496 | .14280 |] .13854 | .15412 
58 | .12510 | .14299 || .13868 | .15431 
59 | .12524 | .14317 |] .13883 | .15451 
60 | .12538 | .14385 || .13897 | .15470 


| 
| 
|| 


47% 


Vers. | Ex. sec. 


80° 
Vers. |Ex.sec.| 
8397 | .15470 
.18412 | .15489 
18427 | .15509 
.18441 | .15528 
.18456 |. .15548 
.18470 | .15567 
.18485 | .15587 
.13499 | .15606 
.13514 | .15626 
.18529 ; .15645 
.18548 | .15665 
.13558_| .15684 
slopioe| .1ov04 
18587 | 115724 
. 18602 ee 
.18616 | .15763 
.13681 | .15782 
.13646 | .15802 
.18660 | .15822 
.18675 | .15841 
.18690 | .15861 
.18705 | .15881 
.18719 | .15901 
.18784 | .15920 
.18749 | .15940 
.18763 | .15960 
1377 . 15980 
.18798 | .16000 
.18808 | .16019 
.18822 | .16089 
.138837 | .16059 
.138852 | .16079 
18 8867 .16099 
£18881 | .16119 
.13896 | .16139 
.138911 | .16159 
.138926 | .16179 
.18941 | .16199 
.18955 | .16219 
.18970 | .16239 
.18985 | .16259 
.14000 | .1627 
.14015 | .16299 
.14030 | .16319 
.14044 | .163839 
-14059 | .16359 
.14074 | .16380 
14089 | .16400 
.14104 | .16420 
14119 | .16440 
.14134 | .16460 
.14149 | .16481 
.14164 | .16501 
AAD) 16521 
14194 | .16541 
.14208 | .16562 
.14223 | .16582 
.14288 | .16602 
.14253 | .16623 
.14268 | .16643 
.14283 | .16663 


.14283 | .16663 
-14298 | .16684 
.14813 | .16704 
~14828 | .16725 
14843 | .16745 
.14858 | .16766 
14373 | .16786 
.14388 | .16806 
.14403 | .16827 
.14418 | .16848 
14433 | .16868 


.14449 | .16889 
14464 | .16909 
14479 | .16930 
.14494 | .16950 
-14509 | .16971 
.14524 | .16992 


145389 | ..17012 
.14554 | .17033 


14569 | .17054 
14584 | .17075 


.14599 | .17095 


-14615 | .17116 


.146380 | .17187 


.14645 | .17158 
.14660 | .17178 
.14675 | .17199 


-14690 | .17220 


14706 | .17241 
14721 | -.17262 
-14786 | 172838 
14751 | .17804 


14766 | .173825 
14782 | .17346 


14797 | .17367 
.14812 | .17388 


14827 | .17409 


.14843 | .17430 
.14858 | .17451 
14873 | 17475 
.14888 | .17493 
.14904 | .17514 
14919 | .17535 
14934 | .17556 
.14949 | .17577 
.14965 | .17598 
.14980 | .17620 
14995 | .17641 
15011 | .17662 
15026 | .17683 
.15041 | .17704 
.15057 | .17726 
15072 | .17747 
.15087 | .17768 
.15103 | .17790 
15118 | .17811 
.15134 | .17832 
.15149 | .17854 
15164 | .17875 
.15180 | .17896 
.15195 | .17918 


— 
—— 


338° 


.16183 


. 19236 


COOBNAMTIR WWE © 


/ 
Vers. |Ex.sec.|| Vers... |Ex. sec. Ex. see.|| Vers. |Ex. sec. 
15195 17918 .16133 | .19236 . 20622 ~ 18085 | 22077 
15211 17939 .16149 | .19259 | 20645 | .18101 | .22102 
15226 17961 .16165 | .19281 .20669 LOIS ji eoalet 
15241 17982 .16181 | .19304 | .20693 || .18185 | .22152 
15257 18004 .16196 | .19327 | .20717 18152. |. 22177 
5272 18025 .16212 | .19349 .20740 .18168 | .22202 
15288 | .18047 || .16228 19372 | ,20764 .18185 | .22227 
15303 | .18068 .16244 | .19894 -20788 || .18202 | .22252 
15319 | .18090 .16260 | .19417 . 20812 .18218 | .22277 
1533 .18111 .1627 .19440 . 20836 18235 | .22802 | 
15350 | .18133 .16292 | .19463 .20859 £18252 |: .22327, | 10 
.15365 | .18155 || .16808 | .19485 .20883 || .18269 | .22352 | tl 
.15381 | .18176 || .16324 | .19508 | .20907 || .18286 | .22377 | 12 
.15396 | .18198 || .16340 | .195381 .20981 |} .18302 | .22402 | 13 
.15412 | .18220 .16355 | .19554 . 20955 .183"9 | .22428 | 14 
.15427 | .18241 .16871 | .19576 | .20979 |; .18886 | .22453 | 15 
15443 | .18263 .16387 | .19599 .21003 .18353 | .22478 | 16 
.15458 | .18285 -16403 | .19622 || .21027 .18869 | .22503 | 17 
.15474 | .18307 .16419 | .19645 || .21051 || .18886 | .2252 / 18 
.15489 | .18328 .16485 | .19668 | .21075 || .18403 | .22554. | 19 
.15505 | .18850 |} .16451 | .19691 | .21099 || .1842 222579 | 20 
15520 | .18372 || .16467 | .19713 || | .21128 || .18437 | .22604 | 21 
.15536 | .18394 || .16483 | .19736 || .21147 || .18454 | .22629 | 22 
.15552 | .18416 || .16499 | .19759 || 21171 || .18470 | .22655 | 23 
615567 | .18437 || .16515 | .19782 || .21195 || .18487 | .22680 | 24 
.15583 | .18459 .16531 | .19805 || sele20 .18504 | .22706 | 25 
.15598 | .18481 -16547 | .19828 .21244 || .18521 | 22731 | 2 
.15614 | .18503 .16563 | .19851 .21268 || .18588 | .22756 | 27 
.156380 | .18525 .16579 | .19874 .21292 || .18555 | .22782 | 28 
.15645 | .18547 || .16595 | .19897 .21316 || £18572 | .22807 | 29 
.15661 | .18569. || .16611 | .19920 21841 || .18588 | .22533 | 30 
.15676 | .18591 .16627 | .19944 .21365 || £18605 | .22858 | 31 
.15692 | .18613 .16644 | .19967 .21389 || .18622 | .22884 | 32 
.15708 | .18635 .16660 | .19990 .21414 .18639 | .22909 | 33 
»15728 | 518657 -16676 | .20013 21438 || 118656 | .22935 | 34 
.15739 | .18679 .16692 | .20036 -21462 .18673 | .22960 | 35 
.15755 | .18701 || .16708 | .20059 .21487 .18690 | .22986 | 36 
.15770 | .18723 .16724 | .20083 ~21511 || 18707 | 28012 | 37 
.15786 | .18745 .16740 | .20106 .21585 .18724 | .23037 | 38 
.15802 | .18767 .16756 | .20129 .21560 .18741 | .23063 | 39 
.15818 | .18790 || .16772 | .20152 .21584 || .18758 | .28089 | 40 
.15833 | .18812-}| .16788 | .20176 .21609 || .18775 | .238114 | 41 
.15849 | .18834 -16805 | .20199 .216383 18792 | .28140 | 42 
.15865 | .18856 .16821 | .20222 | .21658 .18809 | .23166 | 43 
.15880 | .18878 .16837 | .20246 | .21682 18826 | .238192 | 44 
.15896 | .18901 .16853 | .20269 | 21707 .18848 | .28217 | 45 
.15912 | .18923 .16869 | .20292 peliol .18860 | .28243 | 46 
.15923 | .18945 .16885 | .2C316 | .21756 || .18877 | .238269 | 47 
.15943 | .18967 ,16902 | .20389 .21781 .18894 | .238295 | 48 
.15959 | .18990 .16918 | .20363 .21805 18911 | .28321 | 49 
.15975 | .19012 || .16934 | .20386 .21830 || .18928 | .283847 | 50 
15991 | .19034 || .16950 | .20410 .21855 || .18945 | .28373 | 51 
.16006 | .19057 .16966 | .204133 | .21879 -18962 | .28399 | 52 
.16022 | .19079 .16983 | .20457 .21904 .18979 | .23424 | 53 
.16038 | .19102 || .16999 | .20480 .21929 .18996 | .238450 | 54 
.16054 | .19124 || .17015 | .20504 .21953 .19018 | 223476 | 55 
.16070 | .19146 |} .170381 | .20527 -21978 .19030 | .23502 | 56 
216085 | .19169 .17047 | .20551 . 22003 -19047 | .28529 | 57 
.16101 | .19191 .17064 | .20575 . 22028 .19064 | .28555 | 58 
.16117 | .19214 || .17080 | .20598 . 22053 .19081 | .28581 | 59 
.17096 | .20622 22077 .19098 | .23607 | 60 


| 87° | . 88° 3 
, / 
Vers. |Ex.sec. | Vers, |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec. 
0 | .19098 | .23607 20186 | .25214 || .21199 | .26902 ||" .22285 | .28676 | .0 
1 | .19115 | .23638 || .20154 | .25241 || .21217 | ..26981 || .223804 | .28706 1 
2 | .19133 | .23659 || .20171 | .25269 || .21285 | .26960 || .22822 | .28737 | 2 
31 .19150 | .23685 || .20189 | .25296 || .21253 | .26988 || .22340 | .28767 | 3 
4} .19167_| .23711 20207 | .25824 |) .21271 | .27017 || .22859 | .28797 |) 4 
5 | .19184 | .23738 || .20224 | .25351 || .21289 | .27046 |} .22377 | .28828 | 5 
6 | .19201 | .23764 || .20242 | .25379 || .21307 | .27075 || .22895 | .28858 | 6 
~ | (19218 | .23790 || .20259 | .25406 || .21824 | .27104 || .22414 | .2888) | 7 
8 | 19235 | .23816 || .20277 | .25434 |) .21842 | .27138 || .22432 | .28919 | 8 
9 | .19252 | .23843 |! .20294 | .25462 || .213860 | .27162 || .22450 | .28950 | 9 
10 | .19270 | .23869 |} .203812 | .2548 21378 | .27191 || .22469 | .28980 | 10 


19287 
19804 


19321 
19338 


19356 
19873 
,19390 
19407 
,19424 


19442 


19459 


.19476 
.19493 
.19511 
.19528 
.19545 
. 19562 
.19580 
.19597 
.19614 
.19632 
.19649 
.19666 
1684 
19701 
19718 
.19736 
19753 
19770 
19788 | 


.19805 


- 19822 
.19840 


.19857 
19875 
.19892 
.19999 
.19927 
.19944 
.19962 
.19979 
.19997 
.20014 
20082 
20049 
20066 
. 20084 
.20101 
.20119 
. 20136 


.23895 


WY. 
~ 23922 || 


23948 
. 23975 
.24901 
24028 
. 24054 
. 24081 
.24107 
.24134 


24160 | 
24187 

24913 
24240 
24267 
24293 
24320 
24347 
24373 
24400 


AAQ™ 
24427 


24454 
-24481 
.24508 
24534 
.24561 
-24588 
.24615 
. 21642 
.24669 
24696 
s24 (25 
24750 
QATTT 
24804 
24832 
24859 
. 24886 
,21913 
.24940 
24967 
24995 

£25022 

25049 | 
225077 
.25104 


.20347 
20365 
20382 
20400 


20470 


20594 
20612 
.20629 
.20647 
20665 
. 20682 
.20700 

.20718 
20736 
20758 
20071 
20789 


. 20329 


20417 | .25656 
ny~ ~Poce | 
20435 | .25683 | 

| (20453 | .25711 


20488 | .25767 | 
20506 | .25795 
"20523 | 125823 | 
(20541 | 125851 
"20559 | .25879 
(20576 | 125907 | 


20807 
20824 
20842 
.20860 


nQ 
.20878 


20895 
.20913 
20931 
.20949 
20967 
20985 
21002 
-21020 
21038 | 
.21056 
21074 | 
21092 
21109 
21127 
21145 
21163 
.21181 
21199 


ape aty Wf 
.25545 
+L0018 
.25600 
25628 


25739 


25935 
25963 

.25991 
.26019 
26047 


26075 
26104 
26132 
26160 
:26188 
26216 
(26245 
26273 | 


26500. || 


ORO 


26801 
. 263380 


26529 
20557 
26586 
26615 
26643 
26672 
26701 
26729 


Bem 
20787 | 


21396 
21414 
21482 
.21450 
21468 
21486 
21504 
21522 
.21540 
21558 
. 21576 
21595 
21613 
.21631 
.21649 
.21667 
.21685 
21708 
21621 
21739 


221057 


606 


21705 
21794 
.21812 
.21830 
21848 
.21866 
.21884 
21902 
21921 


.26858 || .21939 
26387 || .21957 
26415 || .21975 
26448 || .21993 
26472 || .22012 


22030 
22048 
22066 
22084 
.22103 
eal 
22139 
222157 
22176 
22194 


wouehe 


26815 || -,2223 
26844 || .22249 
26873 | 22267 
26902 || .22285 


efo)779) 


20221 
120200 
evel) 
27308 

21888 
27866 | 
.27296 
27425 


27454 
27483 
627513 

.20042 


OrRroD 


~wHIIA 
.27601 
.27630 
.27660 
.27689 
Peels) 
27748 
226018 


27807 


orRar 


etl O0d 


27267 


#0 

.27896 
24926 
.27956 
.27985 
.28015 
28045 
28075 
228105 
28184 
.28164 
£28194 
28224 
~29204 
28284 
.28314 | 
28344 
28374 

28404 
28484 
28464 


28495 


» 


ee he 


s 20000 
-28585 
.28615 
. 28646 
£28676 


22487 
22006 
99594 


s U, 

. 22542 
. 22561 
22579 
, 22598 
22616 
22634 
.22653 
22671 
. 22690 
22708 
OAH | 
622745 
22764 
-22¢82 
.22801 
-22819 
22838 
. 22856 
22875 
.22893 
.22912 
£22330 
. 22949 
. 22967 
. 22986 
. 28004 
123023 
.23041 
. 23060 
. 23079 
.238097 
.23116 
.23134 
.23153 
23979 
.23190 
$23209 
28298 
.23246 
.23265 
. 238283 
~28002 
523821 
20808 
28358 


oonr 
. 2 wid 


23396 


479 


OU 


.80255 | 5i 
.80287 | 52 
.80818 | 53 
.80350 | 54 


80382 | 55 
.80413 | 56 
.80445 | 57 
80477 | 58 
.80509 | 59 
.80541 | 60 


.29011 | 11 
29042 | 12 
.29072 | 13 
.291038 | 14 
.29188 | 15 
.29164 | 16 
229195 4) Ty 
29226 | 18 
29256 | 19 
.29287 | 20 


29818 | 21 
129349 | 22 
29380 | 23 
120411 | 24 
"29442 | 25 
129473 | 26 
129504 | 27 
29535 | 28 
"29566 | 29 
(29597 | 80 


.29628 i 
.29659 | 32 
.29690 | 33 
£29721 | 34 
.29752 | 85 
.29784 | 36 
.29815 | 37 
29846 | 38 
29877 | 39 
.29909 | 40 


.29940 | 41 
299701 2 
.80003 | 43 
380034 | 44 
.80066 | 45 
.80097 | 46 
.80129 | 47 
.380160 | 48 
.30192 | 49 


30228 | 50 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS., 


| 


Vers. 


203396 
20414 
23433 
23452 
238470 
20489 
28508 
20027 
238545 
. 23564 
-20083 
23602 
23620 
23639 
23658 
28677 
23696 
23014 
23733 
.28152 


oQrr 
roe | 


23790 
23808 
23827 
23846 
23865 
23884 
23903 
23922 
23941 
23959 
23978 
23997 
24016 
24035 
24054 
24073 
24092 
24111 
24130 
24149 


24168 
24187 
24206 
24225 
24244 
24262 
2d281 
.24500 
24320 
24339 
24358 
24377 
24396 
24415, 
24484 
.24453 


24472 


24491 
-24510 
«24529 


40° 


Ex. sec. 


.80541 
. 80573 
.80605 
. 30636 
. 380668 
.80700 
.80732 
.80764 
.80796 
80829 
.30861 


380893 
. 80925 
80957 
. 80989 
.31022 
-81054 
.31086 
.31119 
.d1151 
31183 


81216 
.381248 
81281 
-81313 


81546 | 


31378 
31411 
.31443 
31476 
.381509 
31541 
381574 
.31607 
.31640 
81672 
81705 
31738 
O1T71 
31804 
381837 


.31870 
.51908 
. 31936 
. 381969 
. 82002 
82035 
- 82068 
.82101 
8215: 


.82168 


32201 


99909 
. 8223 


82267 


32801 
82834 


.82868 
.82401 


82434 


. 382468 
82501 


41° 


Vers. 


24529 
24548 
24567 
24586 
24605 
24625 
24644 
24663 
24682 
24701 
24720 
24739 
24759 
2407 

24797 
24816 
24835 
24854 
24874 
24893 
24912 


.24931 


24950 
24970 
24989 
25008 
25027 
25047 
25066 
25085 
25104 
25124 
25143 
25162 
25182 
25201 
25220 
25240 


«20259 


95278 


«On 


20297 


25317 
. 25336 
203856 
253875 
20394 
.20414 
25433 
20452 
25472 
20491 


.20511 
.25530 
25549 
.25569 
-25588 
.25608 
25627 
- 20047 
25666 
.25686 


Ex. sec. 


32501 
82535 
82568 
382602 
- 32636 
82669 
32703 
£82737 
82770 
82804 
328388 
382872 
82905 
382939 
82973 
.83007 
.338041 
83075 
.33109 
.83143 
83177 
838211 
83245 
38327 

33314 
. 33348 
33382 
33416 
-838451 
. 338485 
.33519 
33004 
33098 
83622 
83657 
.33691 
33726 
.33760 
.83795 
33830 
33864 


.33899 
33934 
.383968 
- 840038 
.84088 
84073 
84108 
.84142 
84177 
84212 
84247 
. 84282 
34317 
. 34352 
34387 
.84423 
.34458 
84493 
34528 


84563 


43° 


43° 

Vers. Ex. sec. 
25686 | .34563 
25705 | .84599 
20724 | .34634 
20044 | .34669 | 
.207163 | .84704 
.20783 | .84740 
.20802 | .84775 
.20822 | .384811 
.25841 | .84846 
.20061 | .384€82 
.20880 | .384917 
.25900 | .24953 
.25920 | .84988 
.25939 | .385024 
.25959 | .85060 
.20978 | .85095 
.20998 | .35131 
-26017 | .85167 
26037 | .35203 
-26056 | .85238 
26076 | .3527 
.26096 | .35310 
.26115 | .85346 
.26135 | .385382 
.26154 | .385418 
.26174 | .385454 
.26194 | .385490 
.26213 | .85526 
.26283 | .85562 
.26253 | .85598 
-26272 | .385634 
-26292 | .35670 
26312 | .385707 
.26031 | .85743 
-26351 | 38577 
26871 | .85815 
.26390 | .85852 
.26410 | .35888 
-26430 | .35924 
-26449 | .385961 
26469 | .35997 
.26489 | .386034 
.26509 | .386070 
.26528 | .386107 
.26548 | .386143 
.26568 | .386180 
.26588 | .36217 
»26607 | .86253 
.26627 | .86290 
.26647 | .86327 
.26667 | .36363 
.26686 | .386400 
.26706 | .36437 
26726 | .86474 
.26746 | .386511 
.26766 | .86548 
.26785 | .86585 
-26805 | .386622 
-26825 | .36659 
-26845 | .386696 
.26865 | .38673 


Vers. 


26865 
- 26884 
26904 
.26924 
26944 
26964 


| 26984 
| .27'004 
| 27024 


.24043 
.21063 
-27083 
.21103 
-21128 
27143 
27163 
21183 
21203 
21223 
21243 
21263 
27283 
.27803 
27823 
27843 
21363 
21883 
21403 
21423 
-27443 
21463 
«27483 
27503 
221523 
-27543 
621563 
-27583 
27603 
271623 
271643 
-27663 
-27683 
221103 
£20023 
2771483 
27764 
20784 
27804 
21824 
27844 
27864 
27884 
.27905 
.21925 
20945 
27965 
24985 
28005 
28026 


.28046 
28066 


Ex. sec. 


86788 

86770 
.86807 
86844 
-56881 
.86919 
.86956 


-87293 
.87880 
.87368 
-387406 
.87443 
37481 
387519 


387898 
.37936 
81974 
- 38012 
.38051 
88089 
38127 
38165 
- 88204 
38242 


388280 
38319 
. 88357 
38396 
.38434 
.388473 
38512 
- 388550 
. 88589 
. 88628 
. 38666 
. 88705 
38744 
88783 
. 38822 
. 8&860 
. 38899 
.33938 
88977 


.39016 


| 


45° 


| 


28066 
28086 
.28106 
28127 
28147 
23167 
28187 
28208 
2822! 
28248 
28268 


11 | .28289 
12 | .28309 
13 | .28329 
14 | .28350 
15 | .28370 
16 | .28390 
17-| 28410 
18 | .28431 
19 | .28451 
20 | .28471 
21 | .28492 
22 | .23512 
23 | .28532 
24 | 128553 

25 | .28573 
26 | .28593 
27 | .28614 
28 | .28634 
29 | .28655 
30 | .28675 


—_ pt 
_ Bhan Rees 


31 | .28695 
2 | .28716 


33 | .28786 
34 | .28757 
85 | .28777 
36 | .28797 
87 | .28818 
88 | .28838 
89 | .28859 
40 | .28879 
41 | .28900 
42 | .28920 
43 |. .28941 
44 | .28961 
45 | .28981 
46 | .29002 
47 | .29022 
48 | .29043 
49 | .29063 


50 | .29084 
51! .29104 | 
52 | .29125 | 
53 | 29145 | 
54 | .29166 
55 | .29187 
56 | .29207 
Bz | "99228 
58 | .29248 
59 | 29269 
GO | .29289 


.89685 
39725 


Vers. |Ex. sec. 


.39016 
89053 
89095 
.89134 
.389173 
.89212 
89251 
.39291 
89330 
.39369 
.389409 


89448 
89487 
389527 
39566 
389606 
.39646 


39764 
.dd804 
.39844 
39884 
.39924 
.89963 
.40003 
.40043 
.40083 
.40123 
.40163 
.40203 
.40243 
.40283 
-40324 
.40364 
.40404 
.40414 
.40485 
-40525 
.40565 
.40606 


40646 
40687 
40727 
40768 
.40808 
40849 
40890 
40930 
40971 
41012 
41058 
41098 
41134 
41175 
41216 
41257 
41208 
“41339 
41380 
41421 


| ,29289 
, 29310 


| 120454 


| .29805 


Vers. |Ex.sec.|| Vers. |Ex. sec. 


i 41421 
"41463 


29330 
29351 
29372 
29392 
29413 
29433 


29475 
.29495 
.29516 
£29537 
.29557 
29578 
.29599 
.29619 
.29540 
. 29561 
29581 
.29702 
. 297238 
£29743 
.297'64 
.29785 


29826 
29847 
29868 
.29888 
.29909 
29930 
.29951 
-29971 
29992 
.80013 
80034 
380054 
80075 
.380096 
.30117 


.301388 
.380158 
.30179 
.380200 
~8022 
.80242 
.30263 
.380283 
.30304 
.80825 
.80346 
.80367 
.80388 
00409 
.380430 
.30451 
.80471 
80492 
.30513 
80584 


.42126 
.42168 
. 42210. 
.42251 
42293 
.42335 
42377 
.42419 
.42461. 
-42503 

2545. 
.42587 
.42630 
42672 
42714 
.42756. 
.42799 
.42841 
.42883 
. 42926 
.42968 
.43011 
.43053 
.43096 


438189 
43181 
43224 
43267 
43310 
43352 
43395 
43438 
43481 
43524 
43567 | 
43610 

43653 | 
43596 
43739 
43783 
43826 

43869 
43912 

43956 


481 


46° 
.30534 | .43956 
30555 | .43999 
|| .80576 | .44042 
|| .80597 | .44086 
|| 80618 | .44129 
30639 | .4417% 
|| 80660 | .44217 
| 30681 | .44260 
.30702 | .44304 
-30723 | .44347 
30744 | .44301 
.30765 | .44435 
30786 | .4447 
30907 | .44523 
30828 | .44567 
30849 | .44610 
-80870 | .44654 
.30891 | .44698 
30912 | .44742 
.80988 | .44787 
80954 | .44831 
.809%5 | .44875 
.30096 | .44919 
/31017 | .44963 
.31088 | .45007 
.31059.| .45052 
.31030.| .45096 
.31101 | .45141 
81122 | .45185 
181143 | .45229 
81165 | .4527 
.81186 | .45319 
31207 | .45363 
(31228 | .45408 
(81249 | .45452 
31270 | .45497 
81291 | .45542 
'81312.| .45587 
31384. .45631 
81355 | .45676 
181376 | .45721 
.81397 | .45766 
(31418.| .45811 
| 131489 | .45856 
131461 | .45901 
| 181482 | .45946 
"31503 | .45992 
131524 | .46037 
31545 | .46082 
.81567.| .46127 
181588 | .4617 
.81609 | .46218 
.381630.| .46263 
'31651 | .46309 
| :31673 | .46354 
|| 81694 | .46400 
|| 81715 | .46445 
"31736 | .46491 
"81758 | .46537 
(31779 | .46582 
"31800 | .46628 


| 


.31800 
31821 
81843 
.31864 
81885 
.31907 
31928 
.381949 
31971 
-31992 
82018 


82035 
82056 
82077 
82099 
382120 
32141 
32163 
82184 
82205 
82227 
32248 
82270 
82291 
-32312 
82384 
823595 
82377 
82398 
82420 
82441 


. 82462 
. 32484 
. 32005 
o2027 
.82548 
.o200 

.382091 
| .3826138 
82634 
-382656 
02677 
-32699 
.82120 
02742 
.82163 
~82780 
. 02006 
-82828 
-82849 


82871 


.82893 
82914 


47° 


Vers. 


| .82936 


82957 
82979 
.83001 
80022 
.83044 
.38065 


| 38087 


Ex. sec. 


.46628 
46674 
.46719 
46765 
.46811 
46857 
.46903 
.46949 
.46995 
47041 
47087 
47134 
47180 
47226 
P2772 
47319 
-47365 
47411 
47458 
47504 
47551 
47598 
47644 
47691 


MAIOQ 
. 4 (80 


ATT84 
47831 
. 4787! 

47925 
47972 


48019 


-48066 
.48113 
-48160 
48207 
48254 
.48301 
-48349 
-48396 
.48448 
-48491 


.48538 
48586 
.486383 
48681 
48728 
4877 

48824 
48871 
48919 
-48967 


.49015 
.49063 
49111 
.49159 
49207 
-49255 
.49303 
.49351 
.49399 
49448 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 


ee a a eee 


44° 


oy 
| 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 


Se ee " | 
SS ae 50° ! 51° | 


! | 
Vers. |Ex. sec. Vers. |Ex.sec.!} Vers. |Ex. sec. || Vers. |Ex. sec. 

Sia ae : | = AA | te 
0 | .33087 | .49448 || .343894 | .52495 || 85721 | .55572 {| .87068 | .58902 Oo.) 
1 | .83109 | .49496 384416 | .52476 .385744 | .55626 87091 | .58959 1 
2 | .3381380 | .49544 || .34438 | .59597 || .85766 | .55680 i| .87113 | .59016 2 
8 | .388152 | .49593 .84466 | 252579 .85788 | 55734 .387186 | .59073 3 
4 .383173 | .49641 -384482 | .52630 .85810 | .55789 .387158 | .59130 4 
5 | .83195. | .49690 .84504 | .52681 || .35833 | .55843 .87181 | .59188 5 
6 | .388217 | .49738 .384526 | .52732 85855 | .55897 || .87204 | .59245 6 
7 | .83238. || .49787 .34548 | .52784 385877 | 255951 .87226 | .59302 iy 
8 | .83260 | .49835 .84570 | .52835 .85900 | .56005 .87249 | .59360 8 
9 | .83282 | .49884 .384592 | .52886 .85922 | .56060 87272 | .59418 9 

10 | .33303.| .49933 || .34614 | .52938 || .35944 | . 56114 || .387294 | .59475.| 10 
11 | .33325.| .49981 || .84636 | .52989 |] .35967 | . 56169 || .87317 | .59583 | 11 
12 | .383347 | .50030 .384658 | .53041 .85989 | .56223 .387340 | .59590 | 12 
13 | .38868 | .50079 . 84680 | .53092 .86011 . 56278 .87362 | .59648 | 13 


14 | .33390.| .50128 || .34702 | .53144 || .36034 | _56339 .87385 | .59706 | 14 
15 | .33412 | .50177 || .84724 | .53196 || .36056 | . 56387 || .87408 | .59764 | 15 
16 | .33434 | .50226 || .84746 | .53247 || .36078 -56442 || .87430 | .59822 | 16 
17 | .383455.| .50275 || .84768 | .58299 || .36101 56497 || .387453.| .59880 | 17 
18 | .33477.| .50324 || .84790 | .53351 || .36123 | | 56551 || .87476 | .59938 | 18 
19 | .83499.| .5087% -34812 | .53403 || .386146 | .56606 || .87498 | .59996 19 
20 | .33520.} .50422 || .34834 | .538455 || .36168 .56661 .87521 | .60054 | 20 


21 | .83542 | .50471 .384856 | .53507 |} .86190 | .56716 || .87544 | .60112 21 
22.| .83564.} .50521 84878 | .53559 || .86213 | .56771 .87567 | .60171 | 22 
23 | .33586.| .50570 |} .84900 | .53611 || .36935 | | 56826 || .87589 | .60229 | 23 
24} 33607 | .50619 || .84923 | .53663 || .36258 -56881 .87612 | .60287 | 24 
25 | .83629 | .50669 || .84945 | .53715 || .36980 | - 56987 || .87635 | .60346 | 25 
26 | .33651.} .50718 || .84967 | .53768 || .36802 | | 56992 || .87658 | .60404 | 2 

27 | .83673.| .50767 |] .84989 | 53820 || .36305 | | 57047 || .87680 | .60463 | 27 
28 | .383694./ .50817 || .85011 | .538872 || .36347 -57103 || .87703 | .60521 | 28 
29 | .383716.| .50866 || .85033 |} .53924 || .36370 ~7158 || .37'726 | .60580 | 2 

30 | .38738.| .50916 || .85055 | .53977 || .36392 | 7 57213 || .87749 | .60639 | 30 
31 | .83760.] .50966 || .85077 | .54029 || .36415 | . 57269 || .8777 .60698.| 31 
32 | .33782 | .51015 || .85099 | .54082 || .36¢437 | | 57324 || .87794 | .60756.] 32 
33 | .33803.] .51065 || .35122 | .54134 || .364co0 | | 57380 |} .387817 | .60815 | 3° 

34, .338825 | .51115 || .85144 | .54187 || .36¢482 | | 57436 || .87840.| .60874.| 34 
35 | .83847 | .51165 || .85166 | .54240 || 36504 57491 .37862.| .60983 | 35 
36 | .33869 | .51215 || .85188 | .54292 || /36507 57547 || .87885.| .60992 | 36 
37 | .33891.) .51265 || .85210 | .54345 || .36549 | | 57603 || .87908 | .61051 | 37 
38 | .83912 | .51314 || .35932 | .54398 || .36572 | | 7659 || .87931 | .61111 | 38 
39 | .83934 | .51364.|} .35254 | .54451 || 36594 57715 || .87954.| .61170 | 39 
40 | .33956.] .51415.|| .85277.| .54504 || .36617 | 5777 .37976.| .61229 | 40 


41 | .83978 | .51465 || .85299 | .54557 || .36689 | . 7827 || .87999 | .61288 | 41 
42 | .84000 | .51515 || .3532 -54610 || .386662 | .57883 |] .388022 | .61348 | 42 
43 | .384022 | .51565 || .35343 | .54663 || .36684 | |b? 939 |} .88045.] .61407 | 43 
44°| .34044 | .51615 || .85365 | .54716 || .36707 | | 57995 || .88068 | .61467 | 44 
45 | .384065 | .51665 || .85388 | .54769 || .36799 | | 58051 || .88091 | .61526 | 45 
46 | .34087 | .51716 || .85410 | .54822 || .3¢6759 .58108 4 .38113 | .61586 | 46 

7 | .84109 | .51766 || .85432 | .54876 || 36775 .58164 || .88136 |} .61646 | 47 
48 | .34131 ; .51817 || .85454 | .54929 || .36797 58221 || .88159 | .61705.| 48 
49 | .34153 | .51867 || .85476 | .54982 || .36890 58277 || .38182 | .61765 | 49 

0 | .34175 | .51918 || .85499 | .55086 || .36842 58333 || .38205 | .61825 | 50 

1 | .34197 | .51968 || .85521 | .55089 || . 36865 58390 || .8822 .61885 | 51 
52 | .84219 | .52019 || .85543 | 155143 |) |36887 58447 ||| 388251 | .61945 | 52 

3 | .84241 | .52069 || .85565 | .55196 || .36910 | | 58503 || .88274 | .62005 | 53 
S4 | .34262 | .52120 || .85588 | .55250 || .36932 | “585 30 || .88296 | .62065 | 54 
5d | .34284 | .52171 || .35610 | .55303 || .36955 | | 58617 || .88819 | .62125 | 55 
56 | .34306 | .52222 || .85632 | .55357 || .36978 .58674 || .38342 | .62185 | 56 
57 | .34828 | .52273 || .35654 | .55411 || .37000 .58731 .88365 | .62246 | 57 
58 | .34350 | .52323 || .35677 | .55465 || |aro9¢ -58788 || .38388 | .62306 | 58 

9 | .84372 | .5237 -385699 | .55518 |) .87045 | .58845 || .388411 | _62366 | 59 
60) .34894 | .52425 || .85721 | .55572 || 37068 | | 8902 || .38434 | .62427 | 60 


482 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 


~ 


52° 


53° 


54° 


55° 


Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex. sec. 
| | 
0 | .38434 | .62427 || .89819 | .66164 || .41221 | .70180 || .42642 | .74345 
1 | .38457 | .62487 || .39842 | .66228 || .41245 | .70198 || .42666 | .74417 
2 | 38480 | .62548 || .39865 | .66292 || .41269 | .70267 |) .42690 | .74490 
3 | 38503 | .62609 || .39888 | .66357 || .41292 | .70335 || .42714 | . 74562 
4 | 38526 | .62669 |} .39911 | .66421 || .41316 | .70403 || .42738 | .74635 
5 | .38549 | .62730 || .89935 | .66486 || .41339 | .70472 || .42762 | .74708 
6 | .38571 | .62791 || .89958 | .66550 |} .41863 | .70540 |; .42785 T4781 
7 | 138594 | .62852 || .89981 | .66615 || .41386 | .70609 || .42809 | .74854 
g | 38617 | .62913 || .40005 | 66679 || .41410 | .70677 || .42833 | . 74927 | 
9 | 38640 | .62974 || .40028 | .66744 || .41433 | .70746 || .42857 | 75000 | 
10 | .38663 | .63035 || .40051 | .66809 || .41457 | .70815 || .42881 | .750%3 | 
3866 | .63096 || .40074 | .66873 || .41481 | .70884 || .42905 | .75146 | 
"39709 | .63157 || .40098 | .66938 || .41504 | .70953 || .42029 | .75219 | 
“gare | (63218 || .40121 | .67003 || .41528 | .71022 || .42953 | .75293 | 
38755 | .68279 || .40144 | .67068 || .41551 | .71091 || .42976 | .75366 | 
8877 63341 40168 | .67133 || .41575 | .71160 || 43000 | .75440 | 
88801 | .63402 || .40191 | .67199 || .41599 | .71229 || .48624 | . 75513 | 
"38824 | .63464 || .40214 | .67264 || .41622 | .71298 || .43048 15587 
"99947 | _63525 || .40237 | .67329 || .41646 | .71368 || .48072 | .75661 
| 38870 | .63537 || .40261 | .67394 || .41670 | .71437 || .48096 | .75734 
| “388903 | .63648 || .40284 | .67460 || .41693 | .71506 || .43120 | .75808 | 
38916 | .63710 || .40807 | .67525 || .41717 | .71576 || .43144 | .75882 | 
"38939 | .63772 || .40381 | .67591 || .41740 | .71646 || .43168 | .75956 | 2 
'38962 | .68834 || .40354 | .67656 || .41764 | 71745 || .43192 | .76081 
| /38985 | .63895 || .40378 | .67722 || .41788 | .71785 || .48216 | .76105 | 
"39009 | .63957 || .40401 | .67788 || .41811 | .71855 |) .48240 | .76179 
39032 | .64019 || .40424 | .67853 || .41835 | .71925 || .43264 | .76253 | 2 
"39055 | .64081 || .40448 | .67919 || .41859 | .71995 || .48287 | .76328 | 2 
39078 | .64144 || .40471 | .67985 || .41882 | .72065 |) .43311 | 76402 | 
39101 | .64206 || .40494 | .68051 41906 | .72135 || .48385 | .7647% 
"39124 | .64268 || .40518 | .68117 || .41930 | .72205 || .48359 | .76552 | 
39147 | .64330 || .40541 | .68183 || .41953 | .72275 || .48383 | .76626 | 
"39170 | .64393 || .40565 | .68250 || .41977 | .72846 || .4340¥ | .7 701 
“39193 | .64455 || .40588 | .68316 || .42001 | .72416 || .43431 76776 
39216 | .64518 || .40611 | .68382 || .42024 | .72487 || .43455 | .76851 
"39239 | .64580 || .40635 | .68449 || .42048 | .72557 || .43479 | .76926 | 
39262 | .64643 || .40658! .68515 || .42072 | .72628 || .48503 | .77001 
"39286 | .64705 || .40682 | .68582 || .42096 | .72698 || .48527 | 700% 
"39309 | .64768 || .40705 | .68648 || .42119 | .72769 || .48551 | 77152 
. 39332 | .64831 40728 | .68715 || .42143 | .72840 || .48575 | 77227 
"39355 | .64894 || .40752 | .68782 || .42167 | .72911 || .48599 | .77308 
39378 | .64957 || .40775 | .68848 || .42191 | .72982 || .48623 | .7737 
"39401 | .65020 || .40799 | .68915 || .42214 | .73053 || .48647 | .774 54 
"39494 | 65083 || .40822 | .68982 || .42288 | .73124 || .48671 | .775e 
99447 | .65146 || .40846 | .69049 || .42262 | .73195 || .48695 -77606 
"39471 | .65209 || .40869 | .69116 || .42285 | .73267 || .48720 | .77681 
"39494 | .65272 || .40893 | .69183 || .42309 | .73338 || 48744 | 7757 
"39517 | .65336 || .40916 ; .69250 || .42333 | .73409 || .48768 | .77883 
39540 | .65399 || .40939 | .69318 || .42857 | .73481 | 48792 | . 77910 
"39563 | .65462 || .40963 | .69385 || .42381 | .78552 || 43816 77986 
-39586 | .65526 || .40986 | .69452 || .42404 | .73624 | 43840 | .78062 
39610 | .65589 || .41010 | .69520 || .42428 | .73696 || .48864 | .7818E 
39633 | .65653 || .41033 | .69587 42452 | .78768 || .48888 | .78215 
| 39656 | .65717 || .41057 | .69655 || .42476 |. . 73840 | .48912 | .78291 
39679 | .65780 || .41080 | .69723 42499 | .73911 || .48986 78268 
"39702 | .65844 || .41104 | .69790 || .42523 | .73983 || .48960 78445 
| 39726 | .65908 |} .41127 | .69858 || .42547 | 74056 || .48984 78521 
39749 | .65972 || .41151 | .69926 || .4257 74128 || .44008 | .78598 
"39772 | .66086 || .41174 | .69994 || .42595 | .74200 || .44032 78675 
"39795 | .66100 |} .41198 | .70062 || .42619 | .74272 || .44057 18752 
"39819 | .66164 || .41221 | .70180 Il .42642 | .74345 |! .44081 . 78829 


483 


WODVIATIKRWMH OS 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 


| I 

56° 57° | 58° 59° 
yom | ——w —o 

| Vers. |Ex.sec.|| Vers, |Ex.sec.'| Vers. \Ex. sec.|; Vers. Ex. sec. 

i | ae ey | r " aes near ul 
0 | .44081 | .78829 | .45536 | .83608 -47008 | .88708 || .48496 | .94160 | 
1 | .44105 | .78906 .45560 | .83690 || .47033 | .88796 -48521 | .94254 | 
2 | .44129 | .78984 45585 | .83773 || .47057 | . 88884 |; .48546 | .94349 
3 | .44153 | .79061 45609 | .83855 || .47082 | .88972 || (48571 | .94443 
4 | .44177 | .79138 45634 | .83938 || .47107 | .89060 || .48596 | .94537 | 
5 | .44201 | .'79216 45658 | .84020 || .471381 | .89148 .| .48621 | .94632 | 
6 | .44225 | .79293 45683 | .84103 || .47156 | .89237 || .48646 | -94726 
7 | .44250 | 79871 45707 | .84186 |! .47181 | .89825 || .48671 | .94821 
8 | .44274 | .79449 45731 | .84269 || .47206 | .89414 || .48696 | .94916 
9 | .44298 | .79527 45756 | .84352 || .47230 | .89503 || .48721 | .95011 
10 | .44822 | .79604 45780 | .84435 || .47255 | .89591 || .48746 | .95106 
11 | .44346 | .79682 45805 | .84518 | .47280 | .89680 || .48771 | .95201 
12 | .44370 | .79761 45829 | .84601 || .47304 | .89769 -48796 | .95296 
13 | .44895 ; .79839 45854 | .84685 |! .47329 | .89858 -48821 | .95392 
14 | .44419 | .79917 || .45878 | .84768 47354 | .89948 || .48846 | .95487 | 
15 | .44443 | .79995 45903 | .84852 || .47379 | .90037 |} .48871 | .95583 
16 | .44467 | .80074 45927 | .84935 || .47403 | .90126 || .48896 | .95678 
17 | .44491 | .80152 45951 | .85019 || .47428 | .90216 -48921 | .9577 
18 | .44516 | .80231 45976 | .85103 || .47453 | .90305 -48946 | .95870 
19 | .44540 | .80809 46000 | .85187 || .47478 | .90395 -48971 | .95966 
20 | .44564 | .80388 || .46025 | .85271 || .47502 | .90485 || .48996 | .96062 
21 44588 | .80467 -46049 | .85355 || .47527 | .9057: -49021 | .96158 
22 | .44612 | .80546 46074 | .85489 || .47552 | .90665 -49046 | .96255 
23 | .44637 | .80625 -46098 | .85523 || .47577 | .90755 || .49071 | .96351 
24 | .44661 | .80704 -46123 | .85608 47601 | .90845 || .49096 | .96448 
25 | .44685 | .80783 .46147 | .85692 -47626 | .90935 |} .49121 | .96544 
26 | .44709 | .80862 46172 | .85777 -47651 | .91026 || .49146 | .96641 
27 | .44734 | .80942 .46196 | .85861 -47676 | .91116 || .49171 | .96738 
28 | .44758 | .81021 -46221 | .85946 -47701 | .91207 || .49196 | .96835 
29 | .44782 | .81101 -46246 | .66031 || .47725 | .91297 || .49221 | .96932 
30 | .44806 | .81180 -46270 | .86116 47750 | .91888 || .49246 | .97029 
31 | .44831 | .81260 || .46295 | .86201 || .47775 | .91479 || .49971 | .97197 
32 | .44855 | .81340 -40319 | .86286 .47800 | .9157 -49296 | .97224 
33 44879 | .81419 .46344 | .86871 47825 | .91661 -49321 | .97322 
34 | .44903 | .81499 -46368 | .86457 -47849 | .91752 || .49346 | .97420 
35 | .44928 | .81579 -463893 | .86542 || .4787 . 91844 49372 | 97517 
36 | .44952 | .81659 .46417 | .86627 .47899 | .91935 -49397 | .97615 
37 | .44976 | .81740 -46442 | .86713 || .47924 | .92027 || .49422 | .97713 
38 | .45001 | .81820 -46466 | .86799 || .47949 | .92118 ; .49447 | .97811 
39 | .45025 | .81900 -46491 | .86885 || .47974 | .92210 || .49472 | .97910 
40 | .45049 | .81981 || .46516 | .86990 || .47998 | .92302 |! .49497 | .98008 
41 | .45073 | .82061 || .46540 | .87056 || .48023 | .92394 |! .49522 | .98107 
42 | .45098 | .82142 -46565 |; .87142 -48048 | .92486 .49547 | .98205 
43 | .45122 | .82299 -46589 | .87229 48073 | .92578 || .49572 | .98304 
44 | .45146 | .82303 .46614 | .87315 || .48098 | .92670 || .49597 | .98403 
45 | .45171 | .82384 .46639 | .87401 +48123 | .92762 || .49623 | .98502 
46 | .45195 | .82465 || .46663 | .87488 | .48148 | .92855 .49648 | .98601 
47 | .45219 | .82546 || .46688 | .87574 |} «48172 | .92947 |) .49673 | .98700 
48 | .45244 | .82627 .46712 | .87661 || .48197 | .93040 || .49698 | .98799 | 
49 | .45268 | .82709 46787 | .87748 || .48222 | .93133 .49723 | .98899 
50 | .45292 | .82790 || .46762 | .87834 || .48247 | .93226 || .49748 | .98998 
S1 | .45317 | .82871 || .46786 | .87921 || .48272 | .93319 |) .49773 | .99098 
52 | .45841 | .82953 |} .46811 | .88008 || .48297 | .93412 || .49799 | .99198 
53 | .45865 | .83034 || .46836 | .88095 || .48322 | .93505 || .49824 | .99298 
54 | .45390 | .83116 || .46860 | .88183 || .48347 | .93598 || .49849 | .99398 
55 | .45414 | .83198 46885 | .88270 || .48372 | .93692 || .49874 | .99498 
56 | .45489 | .83280 .46909 | .88357 || .48396 | .93785 || .49899 | .99598 
57 | .45463 | .83362 .46934 | .88445 || .48421 | .93879 || .49924 | .99698 
58 | .45487 | .83444 -46959 | .88532 || .48446 | .93973 .49950 | .99799 
59 | .45512 | .83526 46983 | .88620 |} .48471 | .94066 || .49975 | .99899 
60 |! .45536 | .838608 47008 | .88708 || .48496 | .94160 .50000 '1.00000 


COIATRWEEH OS 


' 


SECANTS., 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL 


i] 


60° | 61° 62° 63° 

(| | | 

Vers. | Ex se || Vers. | Ex. sec.|| Vers,. |Ex. sec. || Vers. | Ex. sec. 
0} .50000 | 1.00000 || .51519 | 1.06267 || .53053 | 1.13005 || .54601 .20269 | 0 
1| .50025 .00101 51544 06375 5307 18122 || .54627 .20395 | 1 
2|. .50050 | 1.00202 || .5157 .06488 || .58104 13239 .54653 20521 | 2 
3| .50076 .00303 .51595 . 06592 .531380 . 13356 .54679 .20647 | 3 
4| .50101 .00404 .51621 .06701 538156 . 18473 .54705 .20773 | 4 
5} .50126 00505 || .51646 | 1.06809 .53181 .18590 || .54731 | 1.20900 | 5 
6| .50151 00607 .51672 .06918 || .58207 .18707 || .54757 .21026 | 6 
7|- .50176 .00708 || .51697 7027 || .58233 .138825 || .54782 21153. | 7 
8} .50202 .00810 || .51723 | M137 || .538258 13942 || .54808 | 1.21280 | 8 
9| .50227 .00912 .51748 .07246 || .538284 .14060 || .54834 .21407 | 9 
10] .50252 01014 |) .5177 .07356 || .53310 .14178 || .54860 .21535 |10 
11| .50277 01116 || .51799 | 1.07465 || .53336 .14296 || .54886 .21662 |11 
12} .50303 .01218 || .51825 | 1.07575 || .53861 .14414 || .54912 .21790 |12 


21918 | 13 
22045 | 14 
22174 | 15 
22302 
22430 |17 
22559 |18 
22688 | 19 
22817 | 20 
22946 
23075. | 22 
23205. | 28 
123334 | 24 
23464 | 25 


14533 || .54938 
.14651 || .54964 
14770 || .54990 
.14889 |!) .55016 
.15008 || .55042 
.15127 || .55068 
15246 || .55094 
15366 || .55120 


15485 ||} .55146 
15605 || .55172 
15725 || .55198 
15845 || .55224 
15965 || .55250 


07685 || .538387 
07795 || .53413 
.07905 || .53439 
-08015 || .53464 
-08126 |} .538490 
.08236 || .53516 | 
08347 || .53542 
08458 || .53567 
C8569 || .58593 
08680 || .53619 
08791 || .58645 
08903. || .53670 
.09014 || .53696 


13} 50328 
14| .50353 
15| .50378 
16] .50404 
17| .50429 
18| .50454 
19| .50479 
20| .50505 


21| .50530 
22| .50555 
23} .50581 
24| .50606 
25} .50631 


-01320 || .51850 

.01422 || .51876 
.01525 || .51901 
.61628 || .51927 

.01730 |) .51952 
.01833 || .51978 
.01936. || .52003 
.02039 || .52029 


.02143 || .52054 
02246 || .52080 
02849 || .52105 
02453 || .52131 
.02557. || .52156 


EA 
ar) 


wo 
pat 


26| .50656 .02661 || .52182 09126 || .58722 16085 || .55276 .23594 | 26 
27| .50682 .02765. || .52207 09238 || .53748 16206 || .55302 23004. | 27 
28} .50707 .02869 || .52233 .09350 |) .5377¢ .16326 || .55828 .20905 | 28 
29| .50732 02973. || .52259 .09462 || .53799 .16447 || .55354 .23985. | 29 
30} .50758 .03077 || .52284 .09574 || .58825 .16568 || .55380 .24116 | 30 
31} .50783 03182 || .52310 .09686 || .53851 .16689 |} .55406 24247 | 31 
32| .50808 03286. || .52385 09799 || .538877 .16810 || .55432 24378 | 32 
33} .50834 .03391. || .52861 .09911. || .53908 16932 || .55458 .24509 | 33 
34) .50859 .03496. || .52386 .10024 || .58928 .17053 || .55484 .24640 | 34 
| 35} .50884 | 1.03601. || .52412 .10137 || .58954 17175 || .55510 -PATI2. | 85 


24903 | 
(25085 | 37 
25167 138 
25300. | 39 


go 
ror 


17297 || .55586 
17419 || .55563 
17541 || .55589 
17663. |) .55615 


1 36| 50910 
| 371 250985 
38| 50960 
39| 50986 


08706. || .52438 
.03811 || .52463 
.038916 || .52489 
.04022 || .52514 


10250 |} .53980 
10363 || .54006 
10477 || .54082 
10590 || .54058 


Pre meh ek ph ph ph eh fre ph fr, fmeh fomeh feel foc focal fore fem fred frm fem fh frre frcmh fom fomech fem fremch fem fm fom fmm fame fame formed fk ponh fomh fou feck fom me ame peed fe frmh frm Pr feck freak fremch fom femh femmh ek femeh fd fk feed peek fed fed 
Pr Fre pre fre face face fre peak mek emer ph eek mh frm fcc famed pam famed — are feed fh free peek meh freak fcc feed frmh — meh fame fd fmm form fom fmm fod frrmh peek fod fom fre fool meh freed fmt fred fmm frm — frceh femd feme fod Leek free peek ped feed bed bad 


Ph bed peek ph peek eek fred peek peak ch ek pk pemeh fed red rek rah feck fk pa eh fh eh fem fra fed fd fred free femme feck fe pe fom famed mh fh fd Pe fee. fom. rk peed rach feck fh feed Pek bed peed feed fel bed beak bed Ped Peed ek Bk 
Pre beh peek frm pened fram fk pak faced fem fh ph fom fem fm fm fom freak free fol fk free peak frm fred prmh foe fk fk fem fed fem fk fom frock pak pk fk fk fh fed fom perk pooh fk fk fk pk ped ph fed feel feed peak Ped pak bod fed pad pod bad 


40| .51011 04128. || .52540 10704 |} .54083 17786 || .55641 . 25482. | 40 
41} .51036 04233 || .52566 .10817 || .54109 .17909 || .55667 . 25565. | 41 
42; .51062 .04339 || .52591 .10981 || .54135 .18031 || .55693 25697. | 42 
43| .51087 | 1.04445 || .52617 .11045 || .54161 18154 || .55719 25880 | 43 
44| .51118 ! 1.04551 || .52642 11159 || .54187 18277 || .55745 .25963 | 44 
5} .51188 | 1.04658 || .52668 11274 || .54213 .18401 || .55771 .26097 | 45 
46| .51163 .04764 || .52694 11888 || .54238 .18524 || .55797 . 26230 | 46 
47| .51189 .04870 || .52719 ; 1.11508 || .54264 .18648 |} .55823 26864. | 47 
48} .51214 04977 || .52745 | 1.11617); .54290 18772. || .55849 .26498 | 48 
49) .51239 | 1.05084 || .5277 11782 || .54816 .18895 || .55876 .26632 | 49 
50; .51265 | 1.05191 || .52796 .11847 || .54342 .19019 || .55902 . 26766 | 50 
151) .54290 ; 1.05298 || .52822 11963 || .54368 .19144 |) .55928 .26900 | 51 
52} .51316 | 1.05405 || .52848 12078 || .54394 .19268 || .55954 .2¢0385 | 52 
53| .51341 .05512 || .52878 .12193.|| .54420 19393 || .55980 27169 |53 
54| .51566 .05619 |} .52899 12809 || .54446 .19517 || .56006 .27804 | 54 
55| .51892 05727 || .52924 12425 || .54471 .19642 || .56082 .27439 | 55 
56| .51417 05835 || .52950 .12540 || .54497 19767 56058 .27574 156 
57| .51443 .05942 |} .52976 12657 || .54523 .19892.|| .56084 27710 | 57 
58| .51468 .06050 || .53001 127% 54549 .20018 || .56111 .27845 | 58 
59| .51494 .06158 || .53027 .12889 || .54575 .20143 || .56137 .27981 | 59 
60! .51519 .06267 || .53053 .18005 |! .54601 .20269 || .56163 .28117 160 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 


/ | ————_————— || — 


Vers. Ex, sec.|| Vers. | Ex. sec.| Vers. | Ex. sec. | Vers. | Ex. sec. 
| ‘ t ! | 
56163 | 1.28117 || .57788 | 1.36620 || .59326 
.56189 | 1.28253 || .57765 | 1.36768 || .59353 
56215, | 1.28390 |, .57791 36916 || .59379 
56241 | 1.28526 || .57817 
1 
1 
1 


64° | 65° i 66° | 67° | 


55930 
56106 | 


56282 


.45859 | .60927 
46020 .| .60954 
46181 | _60980 


| | | 1.37064 || .59406 | 1.46342 | 61007 1.56458 
56267 | 1.28663 || .57844 | 1.37212 || .59433 | 1.46504 | 61034, 1.56634 | 
-56294 | 1.28800 || .57870 | 1.37361 | .59459 | 1.46665 | 61061 | 1.56811 | 


.56320 28937 || .57896 | 
56346 
56372 
.56398 
-56425 | 
HI Hh 11} .56451 
12} .56477 
13} .56503 


.46827 || .61088 

.46989 || .61114 

.47152 || .61141 

.47314 || .61168 

47477 |! 61195 
.47640 || .61222 
47804. |} 61248 
.47967 || 61275 
.48131 || .61302 
.48295 || .61329 
.48459 || .61356 
.48624 || .61383 
.48789 | .61409 
48954 || .61436 
49119 | .61463 


.49284 || .61490 
49450 || .61517 
49616 || .61544 
49782 || .61570 
.49948 .61597 
50115 || .61624 
.50282 || .61651 
50449 | .61678 
50617 | 61705 
41142 || .60125 | 1.50784 || .61732 | 1.61313 |30 


1 

1 

1 

1 

1 

1 

1 56988 
1 
1 
ii 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
i 
1 

41296 || .60152 | 1.50952 || .61759 .61496 | 31 

1 
1 
i 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
di 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
i! 
1 
1 
1 


OlG5) |" 

57342 | 8 | 
57520 
.57698 | 10 
57876 |11 
58054 | 12 
.58233 | 13 
58412 | 14 
58591 | 15 
58771 | 16 
58950 |17 
.59130 |18 
.59311 |19 
59491 | 20 


59672 |21 
.59853 | 22 
.60085 | 23 
.60217 | 24 
.60399 |25 
.60581 | 26 
60763 | 27 
.60946 | 28 
.61129 |2 


29074 || .57928 37658 || .59512 
-29211 || .57949 
.29349 || .57976 


37509 ‘| .59486 

ee 

.29487 || .58002 
| 


37808 || .59539 


ol 
eo) 


“BT957 
38107 
38256 || 


.59566 
59592 


.59619 


ok 
SDOMDVIROAR WMWHS 


-29625 || .58028 


29763 || .58055 38406 || .59645 
88556 |! .59672 


-38707 || .59699 


-29901 ;| .58081 


14| .56529 
15| .56555 
16] .56582 
HE | 17| .56608 
pea 18] 56634 
Wie 19| 56660 

ee 20} .56687 


.30040 || .58108 
.30179 ;| 58134 
.30318 |} .58160 
.30457 || .58187 
.30596 || .58218 
.30735 || .58240 
.30875 || .58266 


.81015 || .58293 
31155 || .58319 
31295 |] .58345 
.31436 || .58372 
.31576 || 58398 
.B1717 || 58425 
.B1858 || .58451 
.31999 || .58478 
82140 || 58504 
32282 || .58531 


1 
i 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
82424 || .58557 | 1 
1 
1 
1 
u 
il 
1 
1 
1 
1 
13 
i i 
i. 
dus 
li 
ET. 
A 
i 
i 
ai 
Te 
aM 
is 
ut 
id 
si 
i% 
1. 
t. 
1. 


38857 || .59725 


39008 || .59752 


39159 |) .5977 


.89311 |) .59805 
89462 |} 59832 
39614 || 59859 
.39766 || .59885 
89918 || .59912 
.40070 || .59988 
40222 || .59965 
40375 || .59992 


21| .56713 
22| .56739 
Hay, 23| .56765 
i 24] .56791 
He 25| .56818 


26] .56844 
27| .56870 
28] .56896 
eae 29] 56923 
30| .56949 


ai 31] .56975 


.40528 |; .60018 
.40681 || .60045 
.40835 || .60072 
.40988 |} .60098 


SSS 


1 
4 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
il 
1 
1 
1 
1 
1 
1 
1 
32| .57001 | 1.382566 || .58584 
i 33] .57028 | 1.32708 || .58610 
ill 34} .57054 | 1.382850 || .58637 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
I 
1 
1 
J 
1 
1 
1 
1 


.51289 || .61812 
.51457 || .61839 
.61866 


41760 || .60232 62049 | 34 
41914 || .60259 
42070 || .60285 
.42295 || .60312 
.42380 || .60339 
.42536 || .60365 
.42692 || .60392 
42848 
43005 
43162 
43318 
43476 

43633 


We | 35| .57080 | 1.32993 || .58663 
Wi 36| .57106 | 1.33185 || .58690 
Wd 37| .57133 | 1.33278 || .58716 
iit 38| .57159 | 1.33422 || 58743 

a 39| .57185 | 1.33565 || .58769 
. 40| .57212 | 1.83708 || .58796 


41] .57288 .83852_ || .58822 
42) 57264 .89996 || .58849 
43} 57291 84140 || .58875 
84284 || .58902 
.84429 || .58928 
.84573 || 58955 


62284 |35 
.62419 | 36 
.62604 | 37 
.62790 | 38 
.62976 | 39 
63162. | 40 


.63348 | 41 
.63585 | 42 
63722 | 43 
.63909 | 44 
.64097 | 45 
64285 


51626 
51795. || .61893 
51965 || .61920 
52134 || .61947 
52304 || .61974 
52474 || .62001 


52645 || .62027 
52815 || .62054 
.52986 || .62081 
53157 
53329 
53500 


“41605 || .60205 61864 |33 


.60419 
.60445 
.60472 
.60499 
. 60526 
. 60552 


.62108 
.62185 
.62162 


7 
oO 


( j 1.84718 || .58981 43790 |! .60579 .53672 || .62189 64473 | 47 | 
48 D7422 ; 1.34863 .59008 43948 || .60606 .538845 || .62216 .64662 | 48 


. 85009 .59034 
.80154 || .59061 


44106 
44264 


.60633 
.60659 


54017 || .62243 
.54190 || .6227 


64851 | 49 | 
65040 | 50 | 


1 
| 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
J 
1 
1 
1 
I 
1 
1 
1 
1 
1 
1 
1 
I 
41450 || .60178 .51120 || .61785 | 1.61680 | 32 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
I 
1 
1 
1 


51| .57501 | -59087 | 1.44428 || .60686 | 1.54363 || .62297 | 1.65229 [51 | 
52) .5Y527 | 59114 | 1.44582 || .60713 | 1.54536 || .62324 | 1.65419 | 52 | 
53) .57554 | | .59140 | 1.44741 || .60740 | 1.54709 || .62351 | 1.65609 |53 
54| 57580 | | .59167 | 1.44900 || .60766 | 1.54883 || .62378 | 1.65799 |54 
55| .57606 | | 59194 | 1.45059 || .60793 | 1.55057 || .62405 | 1.65989 | 55 
56| .57633 | 59220 | 1.45219 || .60820 | 1.55231 |) .62431 | 1.66180 |56 
57| .57659 | 1.36178 || .59247 | 1.45378 || .60847 | 1.55405 || 162458 | 1.66371 [57 
58) .57685 | 1.36325 |) .59273 | 1.45530 || .60873 | 1.55580 || .62485 | 1.66563 | 58 
59) .57712 45699 || .60900 | 1.55755 || .62512 | 1.66755 59 
| 


36473 | .59300 


60) .57738 .86620 |] .59326 66947 


SS 


45859 |! .60927 .55930 !] .62539 30 | 


68° | 69° 70° 
f | / 
Vers. | Ex. sec.|| Vers. | Ex. sec.|| Vers. | Ex. sec.|] Vers. | Ex. sec.) 
0 "62539 1.66947 || .64163 | 1.79043 || 65798 | 1.92380 || .67443 | 2.07155 | 0 
1] .62566 | 1.67139 "64190 | 1.79254 || .65825 | 1.92614 || .67471 | 2.07415 1 
2} .62593 | 1.67332 "64218 | 1.79466 || .65853 | 1.92849 || .67498 | 2.07675 2 
3| .62620 | 1.67525 64245 | 1.79679 || .65880 ee 93083 .67526.| 2.07936 | 3 
4| .62647 | 1.67718 || .64272 | 1.79891 || 65907 | 1.93318 || .67553 | 2.08197 | 4 
5 | 62674 | 1.67911 || .64299 | 1.80104 || .65935 | 1. 93554 67581 | 2.08459 | 5 
6| .62701 | 1.68105 |) .64826 1.80318 || .65962 | 1.93790 || .67608 | 2.08721 | 6 
¢ 62723 | 1.68299 | 64353 | 1.8053 “65989 | 1.94026 || .67636 | 2.08983 | 7 
8} .62755 | 1.68494 || .64381 | 1.80746 || .66017 | 1.94263 || - 7663 | 2.09246 | 8 
9 .62782 1.68689 |; .64408 | 1.80960 66044 | 1.94500 || .67691 | 2.09510 | 9 
10! .62809 | 1.68834 |) .64435 | 1.81175 || .66071 | 1.947837 7718 | 2.09774 | 10 
11} .62836 | 1 69079 .64462 | 1.81390 || .66099 1.94975 || .67746 | 2.10088 | 11 
12| .62863 | 1.69275 || .64489 1.81605 || .66126 | 1.95213 || .67773 | 2.10303 |12 
13 -62390 1.69471 || 64517 1.81821 || .66154 | 1. 95452 .67801 | 2.10568 | 13 
1h yo rend | Syren baer | ee 95691 || .67829 | 2.10834 | 14 
5| .628 69% 6457 82254 || . ) 959¢ 678 : 
16} .62971 | 1.70061 || .64598 | 1.82471 | [66286 | 1. ‘goril ‘Gres 2 11307 16 
: i yee eves ne S a a donee i 96411 || .67911 | 2.11635 |17 
.63025 | 1.70455 |} .0400 82906 || .66290 96652 || .67989 | 2.11903 |18 
Af ee | i | nee ree | oak a 96893 || .67966 | 2.12171 |19 
20) .630% 7085 647 1.83842 || .66345 97135 || .67994 | 2.12440 | 20 
21| .63106 | 1.71050 || .64784 | 1.83561 66373 | 1.97377 || .68021 | 2.12709 |2 
95| .63133 | 1.71249 |) .64761 | 1.83780 |} 66400 | 1.97619 || .68049 | 2.12979 | 22 
23 | -63161 1.71448 || .64789 | 1.83999 66427 | 1.97862 || .68077 | 2.18249 | 23 
24 aie ist || ye Laie 219 || .66455 | 1.98106 .68104 Sera 24 
25) .63215 71847 || .6484¢ 84439 || .66482 | 1.98349 || .68182 18791 | 25 
26 63242 1.72047 64870 | 1.84659 | "66510 | 1.96594 || .68159 | 2.14063 | 26 
ad 63269 1.72247 | . 64898 1.84880 || .66537 | 1.98838 || .68187 | 2.14385 27 
98| 63296 | 1.72448 || .64925 | 1.85102 || .66564 1.99083 || .68214 | 2.14608 | 28 
29| .63323 | 1.72649 || .64952 | 1.85823 || "66592 | 1.99829 |} .68242 | 2.14881 | 29 
30| 63350 | 1.72850 || .64979 | 1.85545 || .66619 1.9957. 68270 | 2.15155 | 3 
31] .63377 | 1.73052 |; .65007 | 1.85767 || .66647 1.99821 || .68297 | 2.15429 | 31 
32! 63404 | 1.73254 || .65084 | 1.85990 || "66674 | 2.00067 || .68825 | 2.15704 | 382 
93| _63431.| 1.73456 || .65061 | 1.86213 || .66702 2.00315 || .68352 | 2.15979 | 33 
34| .63458. | 1.73659 "65088 | 1.86437 || .66729 | 2.00562 || .68 380 | 2.16255 | 34 
35|} .63485 | 1.73862 || .65116 | 1.86661 |) .66756 2.00810 || .68408 | 2.16531 | 35 
36 63512 1.74065 "65143 | 1.86885 || .66784 | 2.01059 || .G8435 2.16808 | 86 
3% .63939 1.74269 || .65170 | 1.87109 || .66811 | 2.01308, .68463 | 2.17085 | 3% 
381 .63566 | 1.74473 || .65197 | 1.873834 "66839 | 2.01557 || .68490 | 2.17863 38 
39| .63594 | 1.74677 || .65225 | 1.87560 "66866 | 2.01807 |} .68518 | 2.17641 | 39 
40 “63621 1.74881 65252 | 1.87785 || .66894 | 2.02057 68546 | 2.17920 | 40 
41} .63648 | 1.75036 6527 1.88011 66921 | 2.02808 || .68573 | 2.18199 | 41 
2| 63675 | 1.75292 || .65806 , 1.88238 "66949 | 2.02559 |} .68601 | 2.18479 | 42 
43) .63702 | 1.75497 || .65834 1.88465 || .66976 | 2.02810 || .68628 2.18759 | 43 
| 44} 63729 1.75703 || .65361 1188692 "87003 | 2.03062 || .68656 | 2.19040 | 44 
45) .63755 | 1.75909 || .65388 | 1.88920 "67031 | 2.03315 || .68684 | 2.19822 | 45 
46) 63783 | 1.76116 || .65416 | 1.89148 "67058 | 2.03568 || .68711 | 2.19604 46 
47| .63310 | 1.76323 .65443. | 1.89376 67086 | 2.08821 68739 | 2.19886 | 47 
48} .63338 | 1.76530 || .65470 | 1.89605 || 67113 | 2.04075 68767 | 2.20169 | 48 
| 49) 63365 | 1.76737 || .65497 | 1.89832 | "67141 | 2.04329 || .68794 | 2.20453 | 49 
50) 63892 | 1.76945 65525 | 1.90063 || "67168 | 2.04584 || .68822 | 2.20737 | 50 
51} .63919 | 1.77154 || .65552 | 1.90293 || .o7196 | 2.04839 || .68849 | 2.21021 | 51 
52! 63946 | 1.77362: | .65579 | 1.90524 || .67223 | 2.05094 |) .688V7 | 2. 21306 | 52 
53| .63973 | 1.77571 || .65607 | 1.90754 |) .67251 | 2.05350 “68905 | 2.21592 |53 
54| .64000 | 1 77780 65634 | 1.90986 || “67278 | 2.05607 || .68932 | 2.21878 | 54 
55| 161027 | 1.77990 | .65661 | 1.91217 || .67306 | 2.05864 || .68960 | 2.22165 | 55 
| 56 .64055 / 1.78200 | .65689 1.91449 || .67333 | 2.06121 || .68988 | 2.22452 | 56 
| 57| .64032 | 1.78410 || .65716 |-1.91681 || .67361 | 2.06379 "9015 | 2.22740 |57 
58| .64109 | 1.78621 65743 | 1.91914 || .67388 | 2.06637 69043 | 2.23028 |58 
59| .64136 | 1.78832 | .65771 | 1.92147 "67416 | 2.06896 || .69071 | 2.23317 |59 
64163 | 1.79043 || .65798 | 1.92380 || “@7443 | 2.07155 || .69098 | 2.28607 | 60 


NATU Ré AL 


TERSED SINES AND EXTERNAL SECANTS. 


71° 


72° 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 


| . 70763 | 


| Vers, | Ix. sec.'| Vers. Ex. sec.'| Vers. | Ex. see 
0; .69098 | 2.28607 10763 2.62796 || .74118 | 2.86370 | 0 
1} .69126 | 2.28897 40791 | 2. 2.63164 . 74146 | 2.86790 | 1 
2| .69154. | 2.24187 . 70818 | 2. 2 68533 74174 | 2.87211.) 2 
3] .6918 2.24478 .10846 | 2. 2.638903 || .74202 | 2.87633 | 3 
4) .69209 | 2.24770 (0874 | 2. 2.64274 || 74231 2.88056 | 4 
5] .69237 | 2.25062 - 70902 | 2. 2.64645 || .74259 | 2.88479 | 5 
6] 69264.| 2.25355 . 70930 | 2.4 2.65018 T4287 | 2.88904 | 6 
7) .69292 | 2.25648 . (0958 | 2.¢ 2.65391 || .74815 | 2.893880 | 7 
8} .69320 | 2.25942 . 70985 | 2. 2.65765 . 74843 | 2.89756 | 8 
| -69347 | 2 26237 || .71013 | 2: 2.66140 || .74371 | 2.90184 
| .69375 | 2.26531 . 71041 | 2.4 2.66515 || .74899 | 2.90613 
.69403 | 2.26827 - 71069 | 2. 2.66892 || .74427 | 2.91042 
.69430. | 2.27123 (1097 | 2.4 2.67269 | (4455 | 2.91473 
.69458 | 2.27420 Ail fe apa le Hs: 2.67647 (4484 | 2.91904 
.69486 | 2.27717 ANG beaks eal! Puhee! 2.68025 . 74512 | 2.92337 
.69514 | 2.28015 Epil tet | allem 2.68405 . 74540 | 2.9277 
.69541 | 2.28313 - 71208 | 2.2 2.68785 || .74568 | 2.93204 
.69569 | 2.28612 41236 | 2.2 2.69167 || .74596 | 2.93640 
.69597 | 2.28912 ~ 61264} 2. 2.69549 . 74624 | 2.94076 
.69624 | 2.29219 . 71292 | 2. 2.69931 . 74652 | 2.94514 
.69652 | 2.29512 -(1820 | 2. 2.70815 - 74680 | 2.94952 
.69680 | 2.29814 +(1348~| "2: 2.70700. || .74709 | 2.95392 
.69708 | 2.30115 BY re WP 2.71085 || . 747387 | 2.95832 
.69785 | 2.30418 (1403, 1-252 2.71471 74765 | 2.96274 
.69763 | 2.30721 oe abba De 2.71858 . 74793 | 2.96716 
.69791 | 2.31024 (1459 | 2. 2.72246 . 74821 | 2.97160 
.69818 | 2.31328 (1487172). 2.72635 . 74849 | 2.97604 
.69846. | 2.31633 Bi gusy teyai ps 2.78024 || .74878 | 2.98050 
.69874 | 2.31939 (1543, | 2. 2.73414 . 74906 | 2.98497 
.69902 | 2.82244 |; .71571 | 2. 2.73806 || .74934 | 2.98944 
-69929 | 2.32551 wh Loose net 2.74198 || .74962 2.99393 
.69957 | 2.32858 . 71626 | 2. 2.74591 || .74990 | 2.998438 
.69985 | 2.33166 (1654 | 2. 2.74984 || .75018 | 3.00293 
- 70013 | 2.33474 MC LOSe les: yeh dao N . 75047 | 3.00745 
. 70040 | 2.33783 a EOS Pay: 2.75775 75075 | 3.01198 
. (0068 | 2.34092 Rdldon ee 2.76171 || .75103 | 8.01652 
.70096 | 2.34403 HAUL AL OH 2.76568 || .75181 | 3.02107 
.10124.| 2.34713 (1794 | 2.5 2.76966 75159 | 8.02563 
. 70151 | 2.35025 SRSA ely ab) 2.773865 | 15187 | 8.038020 
(0179 | 2.85336 ~71850 | 2.2 2.77765 75216 | 3.03479 
- 70207 | 2.35649 PilOre ieee 2.78166 75244 | 3.03938 
- 70235 | 2.35962 ~(1905 | 2. 2.78568 75272 | 8.04398 
. 70263 | 2.36276 41983 | °2..2 2.78970 75800 | 8.04860 
3} .70290 | 2.36590 71961 | 2. 2.79874 (5828 | 8.05822 | 
. 40318 | 2.36905 .71989 | 2.2 2.7977 75856 | 8.05786 
. 10346. | 2.37221 =i BI Wome | | 2.80183 || .75885 | 3.06251 
10874 | 2.37587 72045. | 2.! 2.80589 75413 | 8.06717 
‘| .70401 | 2.37854 (2073. |e eae 2.80996 || .75441 | 8.07184 
| .70429 | 2.88171 72101 | 2.! 2.81404 75469 | 8.07652 
| .70457 | 2.38489 mY he be1) pa Mey | 2.81813 (5497 | 8.08121 
| .40485 | 2.38808 Bi to eal fe Pains 2.82223 75526 | 8.08591 
.705138 | 2.39128 72185 | 2. 2.82633 || .75554 | 3.09063 | 
52! .70540 | 2.89448 PAA PA aSY INB 2p | 2.83045 |! .'75582 | 3.09585 | 
3} .70568 | 2.39768 72241 | 2. | 2.88457 || .75610 | 8.10009 |! 
| 70596 | 2.40089 72269 | 2. 2.83871 || .'75689 | 3.10484 
| .V0624. | 2.40411 (2296. | 2. 2.84285 75667 | 8.10960 
3; .70652 | 2.4073 12024 | 2. 2.84700 75695 | 8.11437 
| .70679 | 2.41057 72352 | 2 2.85116 75723 | 8.11915 
| . 70707 | 2.41381 72880 2.85533 75751 | 3.12394 
9| .70735 | 2.41705 . 72408 2.85951 75780 | 8.12875 
2.420380 72436 2.86370 75808 | 8.13857 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 


76° 
4 
Vers. | Ex. sec. 
0|'.75808 | 3.138357 
1} .75886 | 3.138839 
2] .75864 | 3.14323 
3| .7%75892 | 3.14809 
4| .75921 | 8.15295 
5| .75949 | 3.15782 
6} .75977 | 3.16271 
7|. .76005 | 3.16761 
8) .76034 | 3.17252 
9) .76062 | 3.17744 
10) .76090 | 3.18238 
11| ..76118 | 3.18733 
12} .76147 | 3.19228 
13| «76175 | 3.19725 
14| .76208 | 3.20224 
15) .'762381 | 3.20723 
16) .76260 | 3.21224 
7) .76288 | 3.21726 
18| .76316 | 3.22229 
1 . 763844 | 3.227384 
20| .763873 | 3.23239 
21| .76401 | 3.23746 
22| .76429 | 3.24255 
23| .76458 | 3.24764 
24] .76486 | 3.25275 
25| .76514 | 3.25787 
26| .76542 | 3.26300 
27 | .76571 | 3.26814 
28 .76599 | 3.27380 
29] .76627 | 3.27847 
30| .76655 | 3.28366 
81] .76684 | 3.28885 
32| .76712 | 3.29406 
| 33| .76740 | 3.29929 
34|, .76769 | 3.30452 
385| .76797 | 3.30977 
36} .76825 | 3.81503 
37} .76854 | 3.32031 
38} .76882 | 3.32560 
39] .76910 | 3.38090 
40| .76938°*| 3.338622 
41) .76967 | 3.34154 
42| .76995 | 3.34689 
43) .77023 | 3.385224 
44! .77052 | 3.35761 
45| .77080 | 3.386299 
46| .77108 | 3.36839 
47| .77187 | 3.37380 
48} .77165 | 3.37923 
49) .7719% 8.88466 
50} .77222 | 3.39012 
51} .7%250 | 3.89558 
52] .77278 | 8.40106 
53| .77307 | 3.40656 
54] .773385 | 3.41206 
55) .773863 | 3.41759 
56) .77892 | 3.42312 
57| .77420 | 3.42867 
58 | 77448 | 8.43424 | 
591 .77477-| 3.43982 
77505 | 8.44541 


78° 


. 79209 


17° | 
| 
Vers | Ex. sec.!| Vers. 
| 
77505 | 3.44541 . 79209 
.77583 | 3.45102 .79237 
£77562 | 8.45664 .79266 
.77590 | 3.46228 79294 
.77618 | 3.46793 79823 
77647 | 3.47360 793851 
T1675. | 3.47928 .79380 
.777038 | 8.48498 .79408 
.77782 | 3.49069 .79437 
.77760 | 8.49642 . 79465 
W7788 | 3.50216 || .79493 
T7817 | 8.50791 79522 | 
. 77845 | 3.513868 79550 
V7874 | 3.51947 £1957 
.77902 | 8.52527 || .79607 
77930 | 8.58109 . 79636 
T7959 | 3.538692 . 79664 | 
77987 | 3.54277 .79693 
78015 | 3.54863 79721 
78044 | 8.55451 79750 
78072 | 8.56041 T9778 
78101 | 8.56682 || .79807 
78129 | 3.57224 . 79885 
.78157 | 3.57819 || .79864 
.78186 | 3.58414 || .'79892 
.78214 | 3.59012 || .79921 
. 78242 | 3.59611 79949 
| .78271 | 3.60211 .79978 
. 78299 | 3.60813 .80006 
.783828 | 3.61417 || .80035 
. 78856 | 3.620238 .80063 
. 78884 | 3.626380 .80092 
.78413 | 3.63288 .80120 
.78441 | 3.63849 .80149 
. 78470 | 3.64461 80177 
.78498 | 3.65074 .80206 
.78526 | 3.65690 || .80234 
. 78555 | 3.66807 .80263 
.785838 | 3.66925 .80291 
.78612 | 3.67545 || .803820 
.78640 | 8.68167 || .80848 
. 78669 | 3.68791 80377 
(8697 | 3.69417 .80405 
~ 78725 | 8.70044 80434 
78754 | 3.70673 .80462 
| .78782 | 3.713803 80491 
.78811 | 3.71985 || .80520 
.78839 | 3.72569 || .80548 
.78868 | 3.78205 .80577 
.18896 | 3.78843 . 80605 
. 78924 | 8.74482 .80634 
.78953 | 3.75128 || .80662 
78981 | 3.75766 .80691 
.79010 | 3.76411 |) .80719 
.79038 | 3.77057 || 80748 
79067 | 8.77705 .80776 
.79095 | 3.78355 . 80805 
79123 | 3.79007 . 80833 
.79152 | 3.79661 .80862 
.79180 | 3.80316 80891 
3.80973 .80919 


| 
| 


EX. sec. 


| 3.80973 


3.81633 
3.82294 
3.82956 
3.83621 
3.84288 
3.84956 
3.85627 
3.86299 
3.86973 
3.87649 
3.88327 
8.89007 
3.89689 
3.90373 
8.91058 
3.91746 
3.92436 

93128 

93821 

94517 


wocwmc qwewt) 
DODO 

3) 

ive) 

_— 

os 


17121 
17886 
18652 
19421 
20193 
20966 
21742 
22521 


79° 


| Vers, | Ex. sec. 
| 80919 | 4.24084 
.80948 | 4.24870 
80976 | 4.25658 
.81005 | 4.26448 
| .81083. | 4.27241 
.81062 | 4.280386 
.81090 | 4.28833 
£81119 | 4.29634 
.81148 | 4.30436 
81176 | 4.31241 
.81205 | 4.82049 
| .81233 | 4.82859 
| ,81262 | 4.83671 
.81290 | 4.34486 
.81319 | 4.35804 
.81348 | 4.36124 
.81376 | 4.36947 
.81405 | 4.37772 


.81483 | 4.88600 
.81462 | 4.89430 
.81491 | 4.40263 


|; .81519 | 4.41099 
.81548 | 4.41937 | 


81576 | 4.42778 


| 81605 | 4.43622 |2 
81633 | 4.44468 


.81662 | 4.45317 


| 81691 | 4.46169 
81719 | 4.47023 


.81748 | 4.47881 
.81776 | 4.48740 
.81805 | 4.49603 


.81834 | 4.50468 
.81862 | 4.51337 
.81891 | 4.52208 
.81919 | 4.58081 
.81948 | 4.58958 
81977 | 4.54837 
; 82005 | 4.55720 
| .820384 | 4.56605 
820638 | 4.57493 
82091 | 4.58383 
82120 | 4.59277 
.82148 | 4.60174 
82177 | 4.61073 
.82206 | 4.61976 
.822384 | 4.62881 
82263 | 4.63790 
.82292 | 4.64701 
.82320 | 4.65616 
82349 | 4.66538 
82377 | 4.67454 
82406 | 4.68877 
82485 | 4.69804 
82463 | 4.70284 
82492 | 4.71166 
82521 | 4.72102 
82549 | 4.73041 
82578 | 4.73983 
82607 | 4.74929 
826385 | 4.75877 


— 
Boasorpemre | 


i. 
= 


489 


Vers. | Ex.sec. || Vers. | Ex. see. 
82635 | 4.75877 || .84357 | 5.39245 
.82664 | 4.76829 || .84385 | 5.40422 
82692 | 4.77784 || .84414 | 5.41602 
.82721 | 4.78742 '| .84443 | 5.42787 
82750 | 4.79703 || .84471 | 5.43977 
82778 | 4.80667 || .84500 | 5.45171 
82807 | 4.81635 || .84529 | 5.46369 
82836 | 4.82606 || .84558 | 5.47572 
.82864 | 4.83581 |) .84586 | 5.48779 
.82893 | 4.84558 || .84615 | 5.49991 
82922 | 4.85539 || .84644 | 5.51208 
.82950 | 4.86524 |} .84673 | 5.52429 
82979 | 4.87511 || .84701 | 5.53655 
.83008 | 4.88502 || .84730 | 5.54886 
.83036 | 4.89497 || .84759 | 5.56121 
.83065 | 4.90495 || .84788 | 5.57361 
.83094 | 4.91496 || .84816 | 5.58606 
83122 | 4.92501 || .84845 | 5.59855 
.83151 | 4.93509 || .84874 | 5.61110 
.83180 | 4.94521 || .84903 | 5.62369 
.83208 | 4.95586 || .84931 | 5.63633 
.83237 | 4.96555 || .84960 | 5.64902 
83266 | 4.97577 || .84989 | 5.66176 
.83294:| 4.98603 || .85018 | 5.67454 
.83323 | 4.99633 || .85046 | 5.68738 
83352 | 5.00666 || .85075 | 5.70027 
.83380 | 5.01703 || .85104 | 5.71321 
.83409 | 5.02743 || .85133 | 5.72620 
83438 | 5.03787 || .85162 | 5.73924 
.83467 | 5.04834 || .85190 | 5.75233 
.88495 | 5.05886 || .85219 | 5.76547 
83524 | 5.06941 || .85248 | 5.77866 
.83553 | 5.08000 || .85277 | 5.79191 
.83581 | 5.09062 || .85305 | 5.80521 
.83610 | 5.10129 || .85334 | 5.81856 
.83639: | 5.11199 || .85363 | 5.83196 
83667 | 5.12273 || .85392 | 5.84542 
.83696 | 5.18350 || .85420 | 5.85893 
.83725 | 5.14432 || .85449 | 5.87250 | 
83754 | 5.15517 || .85478 | 5.88612 
.83782 | 5.16607 || .85507 | 5.89979 
.83811 | 5.17700 || .85536 | 5.91352 
83840 | 5.18797 || .85564 | 5.92731 
.83868 | 5.19898 || .85593 | 5.94115 
.83897 | 5.21004 || .85622 | 5.95505 
.83926 | 5.22113 || .85651 | 5.96900 
| .83954 | 5.23226 || .85680 | 5.98801 
| .83983 | 5.24343 || .85708 | 5.99708 
.84012 | 5.25464 || .85737 | 6.01120 
.841041 | 5.26590 |} .85766 | 6.02538 
84069 | 5.27719 || .85795 | 6.08962 
.84093 | 5.28853 || .85823 | 6.05392 
.84127 | 5.29991 || .85852 | 6.06828° 
.84155 | 5.31133 || .85881 | 6.68269 
84184 | 5.32279 || .85910 | 6.09717 
.84213 | 5.33429 || .85939 | 6.11171 
.84242 | 5.34584 || .85967 | 6.12630 | 
.84270 | 5.35743 || .85996 | 6.14096 
84299 | 5.36906 || .86025 | 6.15568 
84328 | 5.38073 || .86054 | 6.17046 
84357 | 5.39245 || .86083 | 6.18530 


490 


Vers. 


82° 


. 86083 
.86112 
.86140 
.86169 
.86198 
.86227 
. 86256 
. 86284 
.86313 
. 86342 
.86371 
-86400 
.86428 
86457 
.86486 
~86515 
-86544 
.86573 
.86601 
. 86630 
.86659 
. 86688 
| .86717 
| .86746 
8677. 

| .86803 
. 86832 
. 86861 
. 86890 
.86919 
.86947 
. 86976 
| .87005 
.87034 
.87063 
-87092 
.87120 
.87149 
87178 
87207 
. 87236 


87265 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 


Ex. sec, || Vers. 
6.18530 || .87813 
6.20020 || .87842 
6.21517 |} .87871 
6.23019 |} .87900 
6.24529 || .87929 
6.26044 || .87957 
6.27566 || .87986 
6.29095 . 88015 
6.30630 . 88044 
6.382171 || .88073 
6.33719 . 88102 
6.35274 || .88131 
6.36835 || .88160 
6.38403 .88188 
6.39978 || .88217 
6.41560 . 88246 
6.43148 88275 
6.44743 || .88304 
6.46346 .88333 
6.47955 . 88362 
6.49571 . 88391 
6.51194 . 88420 
6.52825 . 88448 
6.54462 || .88477 
6.56107 || .88506 
6.57759 || .88535 
6.59418 || .88564 
6.61085 .88593 
6.62759 . 88622 
6.64441 .88651 
6.661380 . 88680 
6.67826 . 88709 
6.69530 || .88737 
6.71242 . 88766 
6.72962 || .88795 
6.74689 || .88824 
6.76424 - 88853 
6.78167 . 88882 
6.79918 .88911 
6.81677 .88940 
6.83443 .88969 
6.85218 .88998 
6.87001 || .89027 
6.88792 || .89055 
6.90592 |} .89084 
6.92400 || .89113 
6.94216 || .89142 
6.96040 || .89171 
6.9787 .89200 
6.99714 |} .89229 
"01565 .89258 

| 7.03423 || .89287 
7.05291 || .89816 
7.07167 || .89845 
4.09052 .893874 
7.10946 .89403 
7.12849 || .89431 
7.14760 || .89460 | 
7.16681 || .89489 
7.18612 || .89518 
% 20551 || .89547 


© WF 3 OTR CO IDE O | 


PIII YWIAI IAI III 


SESE AE AEE eB ge 3 + 


09 00 09 00 G0. 0D 00 G0 G00 OOO -t-3-F-3 ~ 
cS 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 


fat et 
Se DODWIHUPWWHO 


84° 


85° 


Vers. Ex. sec. 
89547 8.56677 
.89576 8.59882 
.89605 8.62002 
.89634 8.64687 
.89663 8.67387 
.89692 8.70103 
.89721 8.728383 
.89750 8.7557 

89779 8.78341 
.89808 8.81119 
.89836 8.83912 
. 89865 8.86722 
,89894 8.89547 
. 89923 8.92389 
.89952 8.95248 
.89981 8.9812 
.90010 9.01015 
.90039 9.03923 
.90068 9.06849 
90097 9.09792 
:9C126 9.12752 
.90155 9.15780 
.90184 9.18725 
.90213 9.21739 
. 90242 9.2477 

. 90271 9.27819 
. 90300 9.30387 
. 90329 9.33973 
. 90358 9.387077 
. 90386 9.40201 
90415 9.43343 
. 90444 9.46505 
90473 9.49685 
. 90502 9.52886 
. 90531 9.56106 
.90560 9.59346 
.90589 9.62605 
.90618 9.65885 
90647 9.69186 
.90676 9.72507 
90705 9.75849 
90734 9.79212 
. 90768 9.82596 
.90792 9.86001 
.90821 9.89428 
90850 9.92877 
.90879 9.96348 
.90908 9.99841 
. 90937 10 .03356 
. 90966 10.06894 
.90995 10.10455 
.91024 10.140389 
~91053 10.17646 
.91082 10212377 
.91111 10.24932 
.91140 10.28610 
.91169 10.32313 
.91197 10.86040 
- 91226 10.39792 
.91255 10.43569 
. 91284 10.47371 


Vers. 


Ex. sec. 


91284 
.91813 
.913842 
91871 
.91400 
-91429 
.91458 
91487 
91516 


91545 © 


91574 


.91603 
.916382 
. 91661 
.91690 
.91719 
91748 
91777 
.91806 
.518385 
. 91864 
.91893 
- 91922 
.91951 
.91980 
. 98009 
. 92038 
. 920587 
. 92096 
.92125 


92154 


92183 
92212 
92241 
92270 
92299 
92328 
92357 
92386 
92415 
92444 
92478 
92502 
92581 
92560 
92589 
.92618 
92647 
.92676 
92705 
927384 
92763 
92792 
92821 
9285 

92879 
92908 
92937 
929366 
.92995 
. 93024 


10.4737 
10.51199 


11.07610 
11.11852 
11 -16125 
11.20427 
11.24761 
11.29125 


11.33521 
11.37948 
11.4408 
11.46900 
11.5142 
1.55982 
11.60572 
11.65197 
11. 69856 
11.4550 
11.%9278 
1184042 
11.88841 
11.9367 
1198549 
12.03458 
1208040 
12. 13388 
12.18411 
12.23472 
12.28572 
12.33712 
12.38891 
12.44112 
12.49373 
12.54676 
12.0021 
12.65408 
2.70038 
16512 


04850 
.10096 
.1588 
.21730 
13.27620 
18:38559 


94766 


86° 
} 
Vers Ex. sec. 

- 93024 13.33559 0 
.93053 13.39547 1 
. 93082 13.45586 2 
.93111 13.51676 3 
.93140 13.57817 4 
.93169 13.64011 5 
. 93198 13 .70258 6 
93227 13.76558 v4 
93257 13.82913 8 
93286. | 13.89323 | 9 
.93315 13.95788 10 
.93844 14.02310 11 
93373 14.08890 12 
. 93402 14.15527 13 
93431 14. 222293 14 
.93460 14. 28979 15 
.93489 14.35795 16 
- 93518 14. 42672 17 
93547 14.49611 18 
98576 14.56614 19 
. 93605 14.68679 | 20 
. 93634 14.70810 | 21 
. 93663 14.78005 2 
93692 14.85268 23 
93721 14.92597 24 
. 938750 14.99995 25 
9377 15.07462 | 2 
. 938808 15.14999 27 
98837 15.22607 |-28 
. 93866 15.30287 | 29 
93895 15.38041 380 
. 93924 15.45869 81 
. 93953 15.58772 | 32 
. 938982 15.61751 33 
94011 15.69808 | 34 
. 94040 15.77944 85 
. 94069 15.86159 36 
. 94098 15.94456 v4 
94127 16.02885 38 
.94156 16.11297 39 
.94186 16.19848 | 40 
94215 16.28476 41 
94244 16.37196 o 
. 94273 16.46005 | 43 
. 94302 16.549038 44 
94331 16.63893 | 45 
.94560 16.72975 | 46 
94889 16.82152 ie 
94418 16.91424 4S} 
94447 1700794 49 
. 94476 17.10262 | 50 
.94505 17.19830 51 
9458 17.29501 52 
. 94563 17.3927 53 
94592 17.49153 | 54 
94621 17.59189 5D 
94650 17.69233 | 56 
.91679 17.79438 57 
94708 17.89755 58 
94-737 18.00185 59 
18.10782 60 


87° 88° 89° 
/ 
Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. 
0 - 94766 18.10732 . 96510 27 65371 - 98255 56. 29869 
1 94795 18 .21397 .96539 2789440 98284 57 .26976 
2 . 94825 18 382182 . 96568 28 18917 98313 |. 58.27431 
3 . 94854 18.43088 . 96597 28.388812 98342 59.31411 _ 
4 .94883 18.54119 . 96626 28 .64137 98371 60.39105 ° 
5 .94912 18. 65275 . 96655 28 .89903 98400 61.50715 
6 .94941 18.76560 . 96684 29.16120 98429 52. 66460 
ff . 94970 18.87976 96714 29 .42802 98458 63. 86572 
8 . 94999 18.99524 .96743 29.69960 98487 65.11304 
9 . 95028 19.11208 96772 29.97607 98517 66 .40927 
10 . 95057 19 .23028 . 96801 80.25758 . 98546 57 . 75736 
11 .95086 19.34989 . 96830 30.54425 .98575 69.16047 
12 .95115 19.47093 . 96859 80.83623 . 98604. 70. 62285 
13 .95144 19.59341 .96888 81.13366 98633 92.14583 
14 .95173 19.71737 .96917 31.48671 98662 3.73586 
15 95202 19.84283 . 96946 81.74554 .98691 75 .39655 
16 . 95231 19.96982 . 96975 382.06030 . 98720 77.1827. 
4 - 95260 20.09838 - 97004 82.38118 .98749 78.94968 
18 . 95289 20. 22852 - 970383 82. 708385 -98778 80.85315 
19 . 95318 20.36027 . 97062 383 .04199 . 98807 82.84947 
20 . 95347 20.49368 .97092 83. 38232 . 98836 84.94561 
2 .95877 2C0.62876 .97121 83. 72952 . 98866 87.14924 | 
22 .95406 20.76555 .97150 84.08380 .98895 89.46886 
2 . 95435 20.90409 .97179 84.44539 . 98924 91. 91387 
2: . 95464 21.04440 . 97208 84.81452 . 98953 9449471 
25 . 95493 21.18653 . 97237 85.19141 . 98982 97 .22303 
1 26 . 95522 21.38050 . 97266 85 .57633 .99011 100.1119 
4 .95551 2147635 97295 85 .96953 .99040 103.1757 
28 . 95580 21.62413 . 97324 36.37127 . 99069 106.4311 
29 . 95609 21 .77386 . 97353 36. 78185 . 99098 109.8966 
i | 30 . 95638 21. 92559 . 97882 87.20155 . 99127 118.5930 
i 95667 22 .07935 97411 387. 68068 .§9156 117.5444 
et 32 . 95696 22 . 28520 .97440 88 .06957 .99186 121.77 
33 . 95725 22 .39316 .97470 88 .51855 . 99215 126 . 382538 
34 . 95754 22553829 .97499 38 .97797 . 99244 131.2223 
35 .95783 22.71563 97528 39 .44820 992738 136.5111 
36 . 95812 22.88022 97557 39 .92963 . $9802 142.2406 
i 37 . 95842 23.04712 - 97586 40 . 42266 - 99331 148 .4684 
Haye | 38 95871 23 .21637 .97615 40. 92772 .99860 | 155.2623 
A 39 -95900 23 . 388802 . 97644 41 .44525 . 99889 162.7083 
40 . 95929 23.56212 .97673 41.97571 .99418 170.8883 
41 . 95958 23.78878 - 97702 42.51961 .99447 179.9350 
42 . 95987 23 .91790 97731 43 .07746 . 99476 189.9868 
43 . 96016 24 .09969 97760 43 .64980 . 99505 201 .2212 
iii 44 - 96045 24.28414 97789 44 23720 . 99585 213.8600 
45 . 96074. 24.47134 97819 44 84026 99564 228 .1839 
: 46 . 96103 24 661382 97848 45 .45963 . 99593 244.5540 
47 . 96132 24.85417 97877 =|. 46.09596 - 99622 263 .4427 
48 . 96161 25 .04994 97906 46 .74997 . 99651 285 .4795 
49 . 96190 25 .24869 . 97985 47 42241 . 99680 311.5230 
50 . 96219 25.45051 . 97964 48 .11406 . 99709 842.7752 
51 . 96248 25 . 65546 .97998 48 82576 . 99738 880. 9723 
52 - 9627 25. 86360 . 98022 49 55840 . 99767 428.7187 
53 . 96307 26 .07503 . 98051 50.31290 . 99796 490.1070 
54 . 96336 26. 28981 .98080 51.09027 . 99825 571.9581 
55 . 96365 26 .50804 .98109 51.89156 . 99855 686.5496 
56 . 96394 26.2978 .98138 52.1790 . 99884 858 .4369 
57 . 96423 26 .95513 .98168 58 .57046 . 99913 1144.916 
58 -96452 7.18417 . 98197 54.45053 . 99942 1717 .874 
59 . 96481 27.41700 . 98226 55.35946 . 99971 3436 .747 
60 .96510 27 .65371 . 98255 56.29869 1.00000 Infinite 


TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 
Ss 


Let pemed 
aa Somyamncwue| 


12 


Le as a Sete ll ee cea 
492 


CUBIC YARDS PER i00 FEET. 


SuOPES 4:1. 


Base | 3ase Base | Base | Base Base Base Base 
12 | 14 16 18 22 | 24 26 28 
45 53 60. We SS) 2b Bet HAS OO. 97 1 £105 
93h ETE 107 122 137 [°° 167 181 196 | 211 
142 | 163 186 208 253 275 297 | 319 
193 | 222 252 281 341 37 400 430 
a4, | 2892 | 319 -| 356 431 468 505 | 542 
300 | 811 | 38) | 433 | 590 | sev | 611 | 656 
356 408 | 460 512 616 668 719 v71 
"415 74 533 593 "11 77 830 889 
475 542 | 608 675 808 875 942 | 1008 
537 611 | 685 %59 907 981 1056 113 
601 682 | 764 845 1008 | 1090 | 1171 1253 
667 756 844 933 1111 4200 | 1289 1378 
734 831 | 926 1023 1216 1312 | 1408 1505 
804 907 1010 1115 1322 1426 | 1530 | 1633 
875 986 1098 | 1208 1431 1542 1653 | 1764 
948 1067 | 1184 | 1304 1541 1659 | 1778 | 1896 
1023 1149 127 1401 1653 1779 | 1905 | 2031 
1100 1233 | 1366 1500 1767 4900 | 20383 | 2167 
1179 | 1319 1460 | 1601 1882 | 2023 | 2164 | 2305 
1259 1407 | 1555 | 1704 | 2000 | 2148 | 2296 | 2444 
1342 1497 1652 1808 2119 | 2275 | 2431 2586 
1426 | 1589 1752 «| «1915 | 2241 2404 | 2567 730 
1512 1682 1853 | 2023 | 2864 | 2534 “05 | 2875 
1600 1778 1955 | 2133 | 2489 | 2667 | 2844 | 3022 
1690 | 1875 2060 | 2245 | 2616 | 2801 | 2986 | 3171 
1781 | 1974 9166 | 2359 | 2744 | 2987 | 38130 | 3822 
1875 2075 9074 o475 | 9875 | 3075 | 8275 | 3475 
1970 | 2178 2384 2593 3007 215 | 3422 | 3630 
2063 | 2282 2496 912 | 3142 | 3356 | 3571 3786 
2167 2389 2610 2833 327 3500 | 3722 | 8944 
2258 2497 2726 2956 | 3416 3645 | 8875 | 4105 
2370 2607 2844 3081 3556 3793 | 4030 | 4267 
2475 | 2719 2964 $208 3697 | 8942 | 4186 | 4431 
2581 | 2833 3085 3337 3841 4093 | 4844 | 4596 
2690 | 2949 3208 3468 | 3986 4245 | 4505 | 4764 
2800 3067 3333 3600 4133 | 4400 | 4667 | 4933 
2912 | 3186 3460 WA | 4282 | 4656 | 4831 | 5105 
3026 | 3307 3589 3870 | 4438 | 4715 | 4996 | 5278 
3142 3431 719 4008 4586 ASTD 5164 | 5453 
3259 | 3556 3852 4148 | 4¢41 5037 | 5333 | 5630 
3379 3682 3986 | 4290 | 4897 201 5505 | 5808 
3500 3811 4422 | 4433 | 5056 | 5367 | 5678 5989 
3623 | 3942 | 4260 4579 | 5216 5534 | 5853 | 6171 
3748 4074 4400 472 5378 | 5704 | 6030 | 6856 
3875 4208 4541 487 B42 «| «5875 6208 6542 
4004 4344 4684. | 5026 | 5707 | 6048 | 6389 | 6730 
4134 4482 | 4830 5179 5875 | 6223 | 6571 6919 
4267 4622 | 4978 | 5833 | 6044 | 6400 "56 | W111 
4401 4764. | 5127 | 5490 | 6216 6579 | 6942 | 7305 
4537 | 4907 | 5278 | 5648 | 6389 6759 | 7130 | '%500 
4675 | 5053 | 5480 | 5808 | 6564 6942 | 7319 | 7697 
4815 | 5200 | 5584 | 59% 6741 7126 | 511 7396 
4956 | 5349 | 5741 6134 | 6919 7312 | 7705 | 8097 
5100 | 5500 | 5900 | 63800 | 7100 | 7500 | 7900 | 8300 
5245 | 5653 | 6060 | 6468 weg2 | 690 | 8097 | 8505 
53903 | 5807 | 6222 | 6637 | 7467 "881 8296 S711 
5542 | 5964 | 6386 | 6808 | %653 | 807 8497 | 8919 
5693 | 6122 | 6552 | 6981 "841 g270 | 8700 | 9130 
5845 | 6282 | 6719 | 7156 8031 8468 | 8905 | 9342 
6000 | 6444 | 6889 | 7333 | 8222 | 8667 | 9111 9556 


Anteater iat EE ts 


TABLE XXX.—CUBIC YARDS PER 100 FEET. 


Depth 
d 


Base 


18 


SLOPES \%: 1. 


Baso 


22 


} Base 


26 


od 
OR OWFH CDOONOUR WOH 


bet et be 


9817 
10096 
* 10380 
; 10667 
han dee te oe OE ee eda ee 
494 


5994 


6222 


6454 
6689 
6928 
7170 
"417 
1667 
7920 
8178 
8439 
8704 
8972 
9244 
9520 
9800 
10083 
10370 
10661 
10956 
11254 
11556 


1861 
2015 
2172 
2333 
2498 
2667 
2839 
3015 
3194 
3378 
8565 
8756 
3950 
4148 
4850 
4556 
4765 
4978 
5194 
5415 
5639 
5867 
6098 
6333 
6572 
6815 
7061 


11506 
11815 
12128 
12444 


Depth 


d 


_ 


~) 


— 
COUR WMD DOCOMO CO? 


beh ed peek Red ek 


a 
yor OTe 
Gre OO 


TABLE XXX.--CUBIC YARDS PER 100 FERT. 


Base | 


14 


14989 
15467 
15952 


16444 


SLOPES 1: 1. 


Base 


30 


115 
237 


367 


Base 


32 


7233 
7585 
7944 
8311 
8685 
9067 
9456 
9852 
10256 
10667 


11085 
11511 
11944 
12385 
12883 
13289 
13752 
14222 
14700 


15185 


| 15678 


16178 
16685 
17200 
17722 
18252 
18789 
19333 
19885 
20444 


TABLE XXX.—CUBIC YARDS PER 


100 FEET. 


SLOPES 14: 1. 


Depth Base Base Base Base Base Base Base Base 
/ d 12 14 16 18 20 28 30 32 
1 49 56 64 71 7 108 116 123 
2 107 122 137 152 167 226 241 256 
3 175 197 219 242 264 353 375 897 
4 252 281 311 341 370 489 519 548 
5 338 3875 412 449 486 634 671 708 
6 433 478 522 567 611 789 833 878 
Fe 538 590 642 694 745 953 1005 1056 
8 652 711 V7 830 889 1126 1185 1244 
9 775 842 908 75 1042 1308 1375 1442 
10 907 981 1056 1130 1204 1500 1574 1648 
11 1049 1131 1212 1294 1375 701 1782 1864 
12 1200 1289 1378 1467 1556 1911 2000 2089 
13 1360 1456 1553 1649 1745 2131 2227 2323 
14 1530 1633 1737 1841 1944 2359 2463 2567 
15 1708 1819 1931 2042 2153 2597 2708 2819 
16 1896 2015 2133 2252 2370 2844 2963 8081 
vi 2094 2219 2345 2471 2597 3101 3227 3353 
18 2300 2433 2567 2700 2833 3367 3500 8633 
19 2516 2656 2797 2938 8079 3642 3782 3923 
I 20 2741 2889 3037 3185 8333 3926 4074 4222 
| 21 2975 31381 8286 3442 8597 4220 437 4531 
22 3219 3381 3544 3707 3870 4522 4685 4848 
93 3471 3642 3812 8982 4153 4834 5005 5175 
24 3733 3911 4089 4267 4444 5156 5333 5511 
il 25 4005 4190 4375 4560 4745 5486 5671 5856 
i 26 4285 4478 4670 4863 5056 5826 6019 6211 
HT 27 457. 477 4975 5175 5375 6175 6375 6575 
Hi 28 4874 5081 5289 5496 5704 6523 - 741 6948 
iy 29 5182 5397 5612 5827 6042 6901 7116 7331 
Hf 30 5500 5722 5944 6167 6389 278 4500 7722 
rt 31 5827 6056 6286 6516 6745 7664 7894 8123 
al 382 6163 6400 6637 6874 vale ial 8059 8296 8533 
if 33 6508 6753 6997 7242 7486 8464 8708 8953 
Hat 34 6863 7118 7367 7619 787 887 9130 9381 
i 385 (227 7486 1745 8005 8264 9301 | . 9560 9819 
th 36 7600 7867 8133 8400 8667 9733 10000 10267 
i 37 7982 8256 8531 8805 9079 10175 =| 10449 | 10723 
nt 38 8374 8656 8937 9219 9500 10626 10907 | 11189 
39 877 9064 9353 9642 9931 11086 | 11375 | 11664 
Vall 40 9185 9481 9778 10074 10370 11556 11852 12148 
| i 41 9605 9908 10212 10516 | 10819 | 12034 | 12338 12642 
Hy 42 10083 10344 | 10656 10967 | 11278 | 12522 12333 13144 
| 43 1047 10790 | 11108 | 11427 | 11745 | 13020 | 13338 | 13656 
44 10919 | 11244 | 11570 11896 12222 | 18526 | 18852 / 14178 
45 11875 11708 | 12042 | 12875 | 12708 14042 | 14875 14708 
} 46 11844 12181 12522 | 12863 | 18204 | 14567 | 14907 | 15248 
47 12316 | 12664 | 13912 | 13360 | 13708 15101 15449 15797 
48 12800 | 13156 | 13511 13867 | 14222 | 15644 16000 16356 
49 18294 13656 14019 14382 | 14745 16197 16569 16923 
50 13796 14167 | 14537 14907: || 15278 | 16759 17130 | 17500 
51 14308 14686 | 15064 15442 15819 | 17331 7708 18086 
52 14830 | 15215 | 15600 15985 16370 | 17911 18296 18681 
53 15360 15753 | 16145 | 16538 16931 18501 18894 | 19236 
54 15900 | 16300 | 16700 | 17100 | 17500 | 19100 | 19500 |- 19900 
55 16449 | 16856 | 17264 17671 18079 | 19708 | 20116 | 20523 
56 17007 17422 | 17837 18252 | 18667 | 20326 | 20741 21156 
57 Lier: 17997 =| 18419 18442 | 19264 | 20953 | 21375 | 21797 
58 18152 18581 19011 19441 19870 | 21589 | 22019 | 22448 
59 18738 19175 19612 .| 20049 | 20486 | 22284 | 22671 23108 
60 19383 1977 23222 | 20667 | 21111 22889 | 23333 | 2377 


496 


© OI OTR OO tO 


Base 


16481 


17094 
17719 
18354 
19000 
19657 
20326 
21006 
21696 
22398 
23111 


Base 


17222 
17850 
18489 
19139 
19800 
20472 
21156 
21850 
22556 
Qa202 


24000 


TABLE XXX.—CUBIC YARDS PER 100 FEET. 


Base 


3133 
8413 
3704 


4005 
4318 
4642 
4978 
53824 
5681 
6050 
6430 
6820 
(222 
7635 
8059 
8494 
8941 
9398 
9867 
10346 
10837 
11339 
11852 


12376 
12911 
13457 
14015 
14583 
15163 
15754 
16356 
16968 
17592 
18228 
18874 
19531 
20200 
20880 
21570 
22272 
22985 
23709 
24444 


Base 


1050 
1244 
1450 
1667 


1894 
2133 
2383 
2644 
2917 
8200 
3494 
3800 
417 
Sid 


4783 
51383 
5494 
5867 
6250 
6644 
7050 
7467 
7894 
8333 


8783 

9244 

9717 
10200 
10694 
11200 
11717 
12244 
12783 
13333 
13894 
14467 
15050 
15644 
16250 
16867 
17494 
18133 
18783 
19444 
20117 
20800 
21494 
22200 
22917 
23644 
24383 
25133 
25894 
26667 


SLOPES 114: 1. 


Base 


8555 


9013 

9482 

9962 
10452 
10954 
11467 
11991 
12526 
13072 
13630 


14198 
1477 
15369 
15970 
16583 
(207 
17843 
18489 
19146 
19815 
20494 
21185 
21887 
22600 
28824 
24059 
24805 
25563 
26332 
ryeae| 


YABLE XXX.—CUBIC YARDS PER 100 FEET, 


SLOPES 2 ; 1. 


es 
Depth | Base | Base | Base | Base | Base | Base |.Base | Base 
d 12 14 16 18 20 28 30 32 
a 52 59 67 74 81 111 119 126 
2 119 133 148 163 178 230 252 267 
3 200 222 244 267 289 378 400 422 
4 296 326 356 385 415 533 563 593 
5 407 444 481 519 556 704 741 78 
6 533 578 622 657 711 889 933 978 
( 674 726 V7 8380 881 1089 1141 1193 
8 830 889 948 1007 1067 1304 1363 1422 
9 1000 1067 1133 1200 1267 1533 1600 1667 
10 1185 1259 1333 1407 1481 1778 1852 1926 
11 1385 1467 1548 1630 1711 2037 2119 2200 
12 1600 1689 177 1857 1956 2311 2400 2489 
13 1830 1926 2022 2119 2215 2600 2696 2793 
14 2074 2178 281 2385 2489 2904 8007 3111 
15 2333 2444 2556 2667 2778 8222 3333 3444 
16 2607 2726 2844 2963 3081 8556 3674 3793 
17 2896 8022 3148 327 3400 3904 4030 4156 
. 18 3200 $323 3457 3600 37383 4267 4400 4533 
i 19 3519 38659 8800 3941 4081 4644 4785 4926 
; 20 3852 4000 4148 4296 4444 5037 5185 5333 - 
Hg 21 4200 4356 4511 4667 4822 5444 5600 5756 
ea 22 45638 730 4889 5052 5215 5867 6030 6193 
23 4941 5114 5281 5452 5622 6304 6474 6644 
24 5333 5511 5689 5867 6044 6756 69338 7111 
25 5741 5926 6111 6296 6481 7222 7407 7593 
ih} 26 6163 6356 6548 6741 6933 7704 7896 8089 
tal vi 6600 6800 7000 7200 7400 8200 8400 8600 
iti 28 7052 (259 7467 VOT4 78381 711 8919 9126 
{ 29 7519 7 3¢ 7948 8163 8378 9237 9452 9667 
Mi 30 8000 8222 8444 8667 8889 9778 10000 10222 
| 31 8496 8726 8956 9185 9415 10333 10563 10793 
ln 32 9007 9244 9481 9719 9956 10904 11141 11378 
i 33 9533 97°78 10022 10267 10511 11489 11733 11978 
i 34 10074 10326 10578 10880 11081 12089 12341 12593 
35 10630 10889 11148 11407 11667 12704 12963 13222 
. 36 11200 11467 11733 12000 12267 13333 13600 13867 
37 11785 12059 12333 12607 12381 13978 14252 14526 
f 38 12385 12667 12948 13280 13511 14637 14919 15200 
i 39 13000 | 13289 | 13578 | 13867 | 14156 | 15311 | 15600 | 15889 
Ha 40 18630 13926 14222 14519 14815 16000 16296 16593 
Hit 41 14274 14578 14881 15185 15 {89 16704 7007 17311 
42 14983 15244 15556 15867 16178 17422 17733 18044 
43 15607 15926 16224 16563 16881 18156 18474 18793 
44 16296 16622 16948 17274 17600 18904 19230 19556 
45 17000 17283 17667 18000 18333 19667 | 20000 | 20383 
1 46 17719 18059 18400 18741 19081 20444 | 20785 | 21126 
47 18452 18800 19148 19496 19844 | 212387 | 21585 | 21933 
48 19200 19556 19911 20267 | 20622 | 22044 | 22400 | 22756 
49 19963 20826 | 20689 21052 | 21415 22867 23230 23593 
50 20741 20711 21481 21852 | 22222 | 23704 24074 | 24444 
51 21288 © | 21911 22289 22667 23044 | 24556 | 24983 | 25311 
52 22341 22726. |.23111 23496 23881 25422 | 25807 26193 
53 23163 | 23556 | 23948 | 24341 24733 | 26304 26696 | 27089 
54 24000 24400 | 24800 25200 25600 27200 27600 28000 
55 24852 | 25259 | 25667 | 26074 | 26481 28111 28519 | 28926 
56 25719 | 261838 | 26548 | 26963 | 27878 | 29037 | 29452 | 29867 
57 26600 | 27022 | 27444 | 27867 | 28289 | 29978 | 30400 | 30822. 
58 27496 27926 | 283856 | 28785 | 29215 | 30933 31363 | 31793 
| 59 28407 28844 | 29281 29719 | 80156 | 31904 | 82341 3277 
60 29333 | 29778 | 30222 | 30667 | 31111 32889 | 338333 | 3377 


498 


| 
| 


Depth 


d 


ODF Oo OTR CO DD 


10 


Base 


12 


56 
133 
933 
856 
500 
667 
856 

1067 
1300 
1556 


1833 
2133 
2456 
2800 
3167 
3556 
3967 
4400 
4856 
5333 


5833 
6356 
6900 
7467 
8056 
8667 
9300 
9956 
10633 
11333 
12056 
12800 
13567 
14356 
15167 
16000 
16856 
17733 
18633 
19556 
20500 
21467 


87395 

88633 
39956 
41300 
42667 


TABLE XXX.—CUBIO YARDS PER 100 FERT. 


Base Base Base Base 
14 16 18 20 
63 7 78 85 
148 163 78 1938 
256 78 800 822 
385 415 444 474 
537 574 611 648 
711 756 800 844 
907 959 1011 1063 
1126 1185 1244 1304 
1367 1433 1500 1567 
1630 1704 1778 1852 
1915 1996 207 2159 
2222 9311 2400 2489 
2552 2648 2744 2841 
2904 3007 8111 8215 
8278 3389 8500 3611 
8674 3793 3911 4030 
4093 4219 4344 447 
4533 4667 4800 4933 
4996 5137 5271 5419 
5481 5630 16 5926 
5989 6144 6300 6456 
6519 6681 8 7007 
7070 7241 411 7581 
7644 7822 8000 8178 
241 8426 8611 8796 
8859 9052 9244 9437 ° 
9500 9700 9900 | 10100 
10163 | 1037 1057¢ 10785 
10848 | 11063 11278 11493 
11556 TA 12000 12222 
12285 12515 12744 12974 
18037 138274 13511 13748 
18811 14056 14300 14544 
14607 14859 | 15111 15°63 
15426 15685 15944 16204 
16267 16533 16800 17067 
17130 | 17404 | 17678 17952 
18015 18296 18578 18859 
18922 19211 19500 19789 
19852 | 20148 | 20444 | 20741 
20804 | 21107 | 21411 Q1715 
2177 22089 | 22400 | 22711 
2277 23093 23411 23730 
237938 | 24119 24444 | 2477 
24833 «| 25167 25500 | 25833 
25896 | 262387 | 26578 | 26919 
26981 27330 |. 27678 |. 28026 
28089 | 28444 | 28800 | 29156 
29219 | 29581 29944 | 30307 
80370 80741 81111 381481 
31544 81922 | 32300 ° | 32678 
82741 33126 838511 33396 
83959 34352 34744 85137 
35200 85600 36000 36400 
36463 36870 8727 37685 
7748 88163 | 8857 38993 
89056 89478 89900 40322 
40385 40815 41244 41674 
41737 42174 42611 43048 
43111 43556 | 44000 | 44444 


30 


22222 
23283 
24267 
25322 


45670 
47111 


Title, 


Number. 


Ratio of circumference to diameter........ 1 4 3.1415927 

Reciprocal Of Same'.........c.scccceeeccecs - 0.3183099 
t 180° 

Degrees in arc of length equal to radius... es 57.295780 
" 

Minutes ‘ hy a ‘ ne _— 8487 .'7468 

Seconds 66 66 66 66 66 ey ae 206264 .81 

a4 Z 

Length of 1° arc, radius unity............... 180° -01745829 

Lengthof 1’ are, “ = “ _7 _ | oooggoso 
tol > @ececeeeeeeeecsese 10800 

Length of 1” are, ‘ Sarah ts ORES f. Pid 000004848 
. 648000 

Radius by which 1 foot of arc = 1 degree. 57 295780 

Radius “ SOT st, TUS “Ss = 1 minute. 343 .'77468 

Radius ‘‘ Sor hs ‘* = 10 seconds 206. 26481 

Factors for dividing a line into extreme 0.6180340 

and ‘mean ratio. . G87. |... HAE in.. Poon 0.3819660 

Base of hyperbolic logarithms.............. € 2. 7182818 

Modulus of common system of logs. = log ¢ M 0.4342945 

Reciprocal of same = hyp. log. 10.......... + 2.3025851 


Length of seconds pendulum at New York 


iInpinches Gcscns 2. baeo. Fees eee C. 8911256 


Length of seconds pendulum at New York 


Tre LeGb.|... ese <.hisee: | Aas} eee 3.25938 
Acceleration due to gravity at New York... g 82.1688 
Square root of same ............0.ecccese pai Vg 5.67175 


Yands in) ] metre, ssc... ees ok Sale cc 


Feet 


1.093623 
3.280869 
89.37043 
0.304797 
0.914892 
1609 .330 


in’ Lacs 


Se eeeeeeosereseeeaseessereseos 


Inchesin 1 ‘* 


Se eeseceee verse sees sso ee Peres 


Metresiin! 12006, .). . acing. «i. < dala iete clue tele s de 


Metres ine diyard 2 cesses bors cetade cieeeee de 


Metres in mile... eis. esic cate. ee 


TABLE XXXI.—USEFUL NUMBERS AND FORMULA, 


0.4971499 | 


9.5028501 | 
1. 7581226 
8.5362739 
5.3144251 
82418774 
6. 4637261 
4.6855749 


1.7581226 
2.5862739 
2.3144251 
9.7910124 
9.5820248 


0.4342945 
9.6377843 
0.3622157 


1.59238162 


0.5131850 
1.5074347 
0. 7537178 


0.0388676 
0.5159889 
1.5951701 
9.4840111 
9.9611324 
3.2066450 


4 


TABLE XXXI.—USEFUL NUMBERS AND FORMULA®. 


i ; Loga- 
Title. Symbol. | Number. | jithm. 
Cubic inches in 1 U.S. gallon.............. 231. 2.3636120 
. «© 86 ‘1 Imperial gallon.......... 277.274 | 2.4429092 
no Semel Ease DUSNOl anne se cores = 2150.42 | 3.33252383 
Cubic feet in 1 U.S. gallon................. 0.133681 | 9.1260683 
* “4 Imperial gallon? .3 4.7. .f- 5 0.160459 | 9.2053655 
= Ks STU USM Clos cs aes cere cae 1.244456 | 0.0949796 
Weight of 1 cub. foot of water, barom. 30 in. 
ther. 39°.88 Fah.; pounds. . 62.379 | 1.'7950384 
se eaOe sy re < ie 62.321 | 1.946349 
Weight in grains, 1 cubic inch, at 62° Fah.. 252.458 | 2.4021892 
No. of grains in 1 pound avoir...........+..- 7000. 3.8450980 
< a LOSI e sp ESAS es apenas 437.5 2.6409781 
Se RO ee ee 
Ege.) 1m0e 
ac=—, 
y = radius of circular arc; r “4 
~t 180° 
1 = length of arc; r= — 
Os) 40 
a° = degrees in same arc. 


Radius by which the length of chord c in feet = a in minutes; 


ea’ 


10 sin 44a’ 


Hyp. log x = com. log x x a or 


com. log (hyp. log x) = com. log (com. log x) + 0.3622157 
Com. log « = M x hyp. log x ; or 


com, log (com. log x) = 9.6377843 + com. log (hyp. log x) 


Circumference of circle (radius = 1)............ceeeeeee seer eeeeees 2r 
MMPESOL CUTIO .. . rasa ss Ledeen nes eo ea a Fae aa ects aine SNe yoo s mr? 
Area of sector (length of arc = 1)......... ee eeee cece rece cer ereece Lélr 
Area of sector (angle Of ATC = A°).....2 ce eee e eee cece eee e eee tenes sap wr? 


Approximate area of segment (chord = c, mid. ord, = HO) ween eee 


APPENDIX, 


Verification of eq. (77). 


a sin f) . 
Hq. (76) p= = aR = sin 7}. cosec W 
sin a 

ay = cos (9. cose : in f) . cot ¥ cosec (76% 

do = ° eae WV - Sin. N 5 > N ((0$) 

do iG . 

ag =P cot 0) — V cot W (7) 
Verification of eq. (81). | 

Differentiating eq. (763) | 
aa } 60 9 6 6 
"FT eee sin  cosee Wo cos 0 cot W cosec W + 


jim 1 6 
ere Det e@ Lf aie ie@3 
We? sin @ cot W cosee WV +- Wa sin @ cosée V 


20 6 p 6 6 
oceaaty « ey aie cot 9 . cot wy a. We cot? V + cosec? Wv 


BRS ( 1 : t E 1 (DP eot2 y 
“an? S478) sn scar cot 9 cot N SP we! cot NV + 1) 


: APPENDIX. 303 


ap” \? 
oie i Sane 
Now (» Z oe 
Pires: ase. == ——— 
ape a’ 
s (3 en ee ee 
pre dj? Pag 


in which substitute for “f. and for ae and let 
t : t a 
cot @ — wv © RG ste a 


(0? + p2(— ay?) 
Oye Penk eta. a 
p? +293 (—a)? — p?(-1- 5s cot 0 cot + + ya 2 cot? 7 +0) 


3 
2)2 
a Ps ue 7 
: 1 0 1 6 1 
1+az+ W CC a ~ Ws cot® W 2? 
La ae . 
2 


1 pe re, { ae 6 
Sa pean 2 et — 4 a< ee sé 
1 an2 t & “+ ; cot N 00d N cot 7) 


(+a)! 


Me nd ( ai Sota) 
5 N2 i hee N 


(+a?) 


1 — syq — acotg 


r) Pre) 


VOEGN | Wl bon OCr 0 KINO, 


53 E. Tenth Street, New York, 
PUBLISH: 


THE RAILROAD SPIRAL. 


The Theory of the Compound Transition Curve reduced to 
Practical Formulz and Rules for Application in Field 
Work, with Complete Tables of Deflections and _Ordinates 
for five hundred Spirals. By Wm. H. Searles, C.E., author 
of * Field Engineering,’ Member of Am. Soc. of C. E. 
Pocket- book form. dth CGiGIOMNs S53 044 bis be oe as 


FIELD ENGINEERING. 


A HAND-BOOK of the Theory and Practice of RAILWAY 
SURVEYING, LOCATION and CONSTRUCTION, designed for 
CLASS-ROOM, FIELD, and OFFICE USE, and containing 
a large number of Usetul Tables, Original and_ Selected. 
By Wm. H. Searles, C.E., late Prof. of Geodesy at Rensseleer 
Polytechnic Inst., Troy. This volume contains many short 
and unique methods of Laying Out, Locating, and Con- 
structing Compound Curves, Side Tracks, and Railroad 
Lines generally. It is also intended as a text-book for 
Scientific Schools. Pocket-book form. 16th edition, 1892. 
1216.4 MIOTOGCO= bos a- ticpra th ee ote: eee oe par gas eet RON 


THE CIVIL ENGINEER’S FIELD-BOOK. 


Designed for the use of the LOCATING ENGINEER. 
Containing Tables of Actual Tangents and Arcs, expressed 
in chords of 600 feet for every minute of intersection, 
from 0° to 96°, from a 1° curve to a 10° curve inclusive. 
Also, Tables of Formule applicable to Railroad Curves and 
the location of Frogs, together with Radii, Long Chords, 
Grades, Tangents, Natural Sines, Natural Versed Sines, 
Natural External Secants, ete. With Explanatory Problems. 
By Edward Butts, C.E. 12mo. 2d edition. Morocco flaps.. 


THE TRANSITION CURVE FIELD-BOOK. 


By Conway R. Howard, C.E. Containing Full Instructions 
for Adjusting and Locating a Curve nearly identical with 
the Cubic Parabola in Transition between any Circular Rail- 
road Curve and Tangent. Simplified in Application by the 
Aid of a General Table, and Illustrated by Rules and Ex- 
amples for various Problems of Location. 12mo, morocco 
HAs hoe eae ES oe LT ATS TY, Bite, STOP oe TRA ed 


TABLES FOR CALCULATING THE CUBIC CON- 


TENTS OF EXCAVATIONS AND EMBANK- 
MENTS BY AN IMPROVED METHOD OF DIAG- 
ONALS AND SIDE TRIANGLES. 
By J. R. Hudson. New edition, with additional tables. 
8v0., GlOoth: 272s... & Li ieih ignad one. oyecere peg" AARBUOET ES el eapaeererepto gs Ae acorn 
METHOD OF CALCULATING THE CUBIC CON- 
TENTS OF EXCAVATIONS AND EMBANKE- 
MENTS BY THE AID OF DIAGRAMS, 


Together with Directions for Estimating the Cost of Earth- 
work. By John C. Trautwine, C.E. Ninth edition, revised 
and enlarged by J. C. Trautwine, Jr. 8vo, cloth..... .....-. 


. $1 SC 


3 00 


2 50 


1 00 


CIVIL ENGINEER’S POCKET-BOOK 


Of Mensuration, Trigonometry, Surveying, Hydraulics, 
Hydrostatics, Instruments and their adjustments, Strength 
of Materials, Masonry, Principles of Wooden and [ron Root 
and Bridge Trusses, Stone Bridges and Culverts, Trestles, 
Pillars, Suspension Bridges, Dams, Railroads, Turnouts, 
Turning Platforms, Water Stations, Cost. of Earthwork, 
Foundations, Retaining Walls, etc. In addition to which 
the elucidation of certain important Principles of Construc- 
tion is made in a more simple manner than heretofore. 
By J. C. Trautwine, C.E. 12mo, morocco flaps, gilt edges. 
41st thousand, revised and enlarged, with new illustrations, 
DYuJ;,. Coe ra ittwine,- bh Gy bn. eon: a cerech pe reel eeet eee 


THE FIELD PRACTICE OF LAYING OUT CIRCULAR 
CURVES FOR RAILROADS. 
By J. C. Trautwine, Civil Engineer. 18th edition, revised 


by J. C. Trautwine, Jr. 12mo, limp morocco. ............... 
THE ECONOMIC THEORY OF THE LOCATION OF 
RAILWAYS. 


An Analysis of the Conditions controlling the laying out 
of Railways to effect the most judicious expenditure of 
capital. By Arthur M. Wellington, Chief Engineer of the 
Vera Cruz and Mexico Railway, etc. New and improved 
edition. 8yvo 


Ceo eee oo e Ose BOF oes es Fees Cer eoeseererser oeeerteen ses 


A TREATISE UPON CABLE OR ROPE TRACTION. 


As applied to the working of STREET and other RAIL+ 
WAYS. (Revised and enlarged from Engineering.) By J. 


A TREATISE ON CIVIL ENGINEERING. 


By D. H. Mahan. Revised and edited, with additions and 
new plates, by Prof. De Volson Wood. With an Appendix 
and complete Index. New edition, with chapter on River 
Improvements, by F. A. Mahan. 8vo, cloth 


C4 wie are le 8 6 ahere-e) aves 


RAILROAD ENGINEERS’ FIELD-BOOK AND EX- 
PLORERS’ GUIDE. 
Especially adapted to the use of Railroad ‘Engineers, on 
LOCATION and CONSTRUCTION, and to the Needs of 
the Explorer in making EXPLORATORY SURVEYS. By 
H.C. Godwin. 2d edition. 12mo, morocco flap 


CC ee 


ENGINEERS’ SURVEYING INSTRUMENTS. 


By Ira O. Baker. 2d edition, revised and greatly enlarged. 
Bound in cloth. 400 pages. 86 illustrations. Index. 12mo, 
GOGH ype AEST sidicy «Enso EE es ca ee 


MANUAL OF IRRIGATION ENGINEERING. 
By Herbert M. Wilson, C.E. Part I. HYDROGRAPHY. 
Part II. CANALS and CANAL WORKS. Part III. STOR- 
AGH RESHRVOLRS 8vVO Clothes teeta . ee een eee 


HIGHWAY CONSTRUCTION. 
Designed as a Text-Book and Work of Reference for all 
who may be engaged in the Location, Construction or Main- 
tenance of Roads, Streets and Pavements. By Austin T. 
Byrne, C.E.. 8vyopelothycnssh: 2) SAL eee ee. ee ee 


THE TRANSITION CURVE. 
By Professor Charles L. Crandall. 12mo, mor. flap... ....... 


2 50 


5 00. 


2 50 


5 00 


2 50 


3 00 


4 00 


a RN a 


‘Seeiestd 


ding fee Gini eet 


ere er 


weno 


peg hs pity 


Oe be le Pp be pa ee Pe Le n= eee tele 


i 


! 


| 
34 


9 


75 
9 UI 1a vusgouVvIuY 


7 


wi 


| 


———— 
——= 
— 
rem 
=— 
—————— 
Sr 
— 
———————— 
a 
————_ 
——— 
=——— 
See 
———— 
— 
rs 
—— 


e2 ar, re 
Fale a eat BRS Sn eS fe tina AS acme ine es eee i" t 


? <I ee : ‘ ‘ a : Bite , . 
aaa Meee caetnetay” | : RR eh EET AR 


